Algebra & Trigonometry (10th Edition)

Chapter 12

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3, 5, 7, 9 ; The term is .Substitute in and ... more

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1, ; The term is Substitute in and obtain the ... more

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6, 7, 8, 9, 10 ; The terms are ,, where . ... more

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1, 3, 4, 7, 11 ; The terms are , , and , where . ... more

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2, 4, 5, , ; The term is . Substitute in and ... more

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; Simplify the expression using the definition ... more

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; Obtain the sum .   ; more

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Infinite sequence ; A function or a collection of ... more

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Finite ; A function or a collection of elements ... more

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Recursively ; A function or a collection of ... more

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Factorial ; Let  be any positive integer. Then,  ... more

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Index of summation ; The sum of the first terms is... more

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Series ; The sum of terms of finite or infinite ... more

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-3, 1, 5, 9, 13 ; The term is . Substitute in ... more

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5,3,5,3,5 ; Substitute 1 and 2 for in to find ... more

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2,0,2,0,2 ; Substitute 1 and 2 for in to find ... more

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-2, 4, -8, 16, -32 ; Substitute 1 and 2 for in ... more

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; Substitute 1 and 2 for in to find the first ... more

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; The function is independent of . This implies ... more

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6,-6,6,-6,6 ; Substitute 1 and 2 for in to find ... more

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,9, ; Substitute 1 and 2 for in to find the ... more

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1, ; Substitute 1 and 2 for in to find the first... more

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; Substitute 1 and 2 for in to find the first ... more

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3, ; Substitute 1 and 2 for in to find the first... more

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0, 0, 6, 24, 60 ; Substitute 1 and 2 for in to ... more

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-5, -4, 9, 40, 95 ; Substitute 1 and 2 for in to... more

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; Substitute 1 and 2 for in to find the first ... more

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; Substitute 1 and 2 for in to find the first ... more

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-73 ; It is observed that . Substitute 25 for in... more

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-240 ; It is observed that . Substitute 16 for in... more

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. ; It is observed that . Substitute 11 for in ... more

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; It is observed that . Substitute 13 for in to... more

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Perform the following operations on the graphing ... more

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Perform the following operations on the graphing ... more

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Perform the following operations on the graphing ... more

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Perform the following operations on the graphing ... more

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Perform the following operations on the graphing ... more

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Perform the following operations on the graphing ... more

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Find the first and tenth term of the sequence by ... more

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; Write the values of and the corresponding ... more

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; The term of the sequence is 1 less than the ... more

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; The term of the sequence is 2 more than the ... more

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; Each term of the sequence is 9 less than the ... more

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; The odd terms of the sequence are 1 and the ... more

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; Each term of the sequence is either 1 or 3, ... more

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; Keep 1 and 2 constant in the numerator and ... more

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; The denominator is increasing in the powers of ... more

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; The numerator is 1 more than the previous term... more

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; The numerator is a series of numbers with the ... more

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; The denominators of the terms of the sequence ... more

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; The terms of the sequence are increasing ... more

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; The numerators are in the form of powers of 3 ... more

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; The numerators of the sequence are increasing ... more

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28, 24, 20, 16, 12 ; Use the recursive relation to... more

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3,4,6,10,18 ; Let ,,,, and be the first five ... more

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81,27,9,3,1 ; Let ,,,, and be the first five ... more

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14, -28, 56, -112, and 224 ; Use the recursive ... more

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1, 2, 2, 3, and ; The initial five terms of any ... more

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-1, 1, 0, 1, 1 ; The initial five terms of any ... more

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1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and 1441, 2... more

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Use the relation and write the relation for . Now... more

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5, 5, , , and ; Expand to find the first five ... more

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1, , , , and ; Expand to find the first five ... more

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6, -24, 60, -120, and 210 ; Expand to find the ... more

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-1,,, and ; Expand to find the first five terms ... more

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; If there is any positive integer then the ... more

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495 ; If there is any positive integer then the ... more

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; If there is any positive integer then the ... more

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; If there is any positive integer then the ... more

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90 ; For any sequence, the sum of first terms can ... more

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40. ; For any sequence, the sum of first terms ... more

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. ; For any sequence, the sum of first terms can ... more

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25. ; For any sequence, the sum of the first ... more

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88. ; For any sequence, the sum of the first ... more

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238. ; Apply property of sum. ; Simplify the ... more

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. ; Simplify the expression by substituting values... more

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. ; Simplify the expression by substituting values... more

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; Use the summation functionality of the graphing... more

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; Use the summation functionality of the graphing... more

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1.33 ; Use the summation functionality of the ... more

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4468.625 ; Use the summation functionality of the ... more

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. ; Let the index of the summation be .The lower ... more

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. ; Let the index of the summation be .The lower ... more

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. ; Let the index of the summation be .The lower ... more

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. ; Let the index of the summation be .The lower ... more

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; Let the index of the summation be .The lower ... more

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; Let the index of the summation be .The lower ... more

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; Let the index of the summation be .The lower ... more

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; Let the index of the summation be .The lower ... more

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; Let the index of the summation be .The lower ... more

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; To write the sum in the summation notation, ... more

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; To find the desired sum, expand the summation ... more

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; To find the desired sum, expand the summation ... more

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. ; The sum is calculated by expanding the terms ... more

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. ; The sum is calculated expanding the terms of ... more

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True. The summation function follows the additive ... more

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True. The terms of the sequence on both the sides ... more

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; The number of accounts .  Substitute... more

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Consider the expression and use the properties of... more

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Consider the expression . Apply the formula and ... more

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; The second term in the expansion should beThus... more

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The first term is missing. ; The total term after... more

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; ; ; ; 3 ; Let the term of the arithmetic ... more

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; For an arithmetic sequence with term as and ... more

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; The term of the arithmetic sequence can be ... more

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79 ; The and terms of the arithmetic sequence ... more

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217 ; Total number of terms in the sequence is 7 ... more

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43,560 ; The and terms of the arithmetic ... more

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1470 ; The and terms of the arithmetic sequence ... more

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2500000 ; The annual sales form an arithmetic ... more

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arithmetic ; In an arithmetic sequence, the ... more

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; For an arithmetic sequence with the first term... more

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recursion ; For an arithmetic sequence with the ... more

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finite arithmetic sequence ; For a finite ... more

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No ; Each term of the sequence is double the ... more

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Yes;5 ; The difference between any two consecutive... more

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Yes;-2 ; The difference between any two ... more

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No ; Each term of the sequence is half the ... more

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Yes; ; The difference between any two consecutive ... more

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Yes;-0.7 ; The difference between any two ... more

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No ; A sequence of square terms cannot be ... more

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Yes; ; The value of is 0, is approximately 0.693... more

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8, 11, 14, 17, 20;Arithmetic sequence;3 ; ... more

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97, 94, 91, 88, 85;Arithmetic sequence;-3 ; ... more

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7, 3, -1, -5, -9; Arithmetic sequence;-4 ; ... more

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1, 3, 7, 13, 21;Not an arithmetic sequence ; ... more

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-1, 1, -1, 1, -1;Not an arithmetic sequence ; ... more

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2,1, 4, 3, 6;Not an arithmetic sequence ; ... more

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2, 8, 24, 64, 160;Not an arithmetic sequence ; ... more

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 Not arithmetic. ; To find the first five terms of... more

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; The formula for the term is of the form , ... more

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; The formulae for the term is of the form , ... more

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; The formula for the term is of the form , ... more

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; The formula for the term is of the form , ... more

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; Calculate the common difference between any ... more

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; Calculate the common difference between any ... more

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. ; The term of the arithmetic sequence is . ... more

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; Substitute and in the general equation of the... more

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; Substitute in the general equation for the ... more

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; Substitute in the general equation for the ... more

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; The term of an arithmetic sequence can be ... more

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; The term of an arithmetic sequence can be ... more

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; Use the values of and to calculate the common... more

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; Substitute in the general equation for the ... more

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; Substitute in the general equation for the ... more

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; Substitute in the general equation for the ... more

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; Determine the common difference by comparing ... more

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; Determine the common difference by comparing ... more

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. ; Determine the common difference by comparing ... more

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; Find the term using the recursion formula .... more

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-49 ; Substitute the value of in the recursion ... more

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83 ; Substitute the value of in the recursion ... more

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; Substitute the value of in the recursion ... more

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-92.4 ; Substitute the value of in the recursion ... more

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110 ; Substitute the number of terms in the ... more

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70. ; Substitute the number of terms in the ... more

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-25 ; Substitute number of terms in the sequence... more

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0 ; Substitute number of terms in the sequence, , ... more

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10000. ; Substitute the number of terms in the ... more

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-4550 ; Substitute number of terms in the sequence... more

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15100 ; Substitute the value of in the recursion ... more

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19200 ; Substitute the value of in the recursion ... more

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-7020 ; Substitute the value of in the recursion ... more

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375 ; Substitute the value of in the recursion ... more

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1275 ; The partial sum of a finite arithmetic ... more

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26425 ; The partial sum of a finite arithmetic ... more

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129250 ; The partial sum of a finite arithmetic ... more

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218625 ; The partial sum of a finite arithmetic ... more

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-28300 ; The partial sum of a finite arithmetic ... more

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33525 ; The partial sum of a finite arithmetic ... more

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The graph of and is a straight line and the ... more

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Find the values of to from the given line ... more

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Find the values of to from the given line ... more

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Find the values of to from the given line ... more

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Find the values of to    from the given line ... more

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2430 ; The arithmetic sequence is with the common... more

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203. ; The first row of wall has one-half brick ... more

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784 ; The falling pattern of the object is an ... more

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; To find the total sales during the first , find... more

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The terms of this sequence contain the first term ... more

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True.Use the equation , where is the difference ... more

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True.The partial sum for the nth series could be ... more

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Use the first two terms to find the common ... more

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; The nth term of the sequence implies , where ... more

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. ; The nth term of sequence implies where is ... more

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The errors are and . ; The sum of the first 50 ... more

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-12, 24, -48, 96.-2. ; A geometric sequence has an... more

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2, 8, 32, 128, 512 ; If a geometric sequence has ... more

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104.2 ; Find the term of a geometric sequence ... more

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; Determine the common ratio using the formula . ... more

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; Express the relation between and . ; Compute ... more

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2.6664 ; Express the term as the sum of individual... more

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; Use the formulae for compound interest , where... more

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Geometric. ; The sequence is geometric when there... more

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; Since the common ratio of the geometric ... more

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; The sum of a finite geometric sequence   which ... more

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Geometric series. ; The sum of an infinite ... more

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Geometric; ; A sequence is geometric if the ratio ... more

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Not geometric. ; A sequence is geometric if the ... more

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Geometric; . ; A sequence is geometric if the ... more

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Geometric; . ; A sequence is geometric if the ... more

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Not geometric. ; A sequence is geometric, if the ... more

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Geometric; ; A sequence is geometric if the ratio ... more

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Not geometric. ; A sequence is geometric if the ... more

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Geometric; . ; A sequence is geometric if the ... more

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  ; Apply the formula of term, i.e., to find the... more

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; Apply the formula of term, i.e., to find the ... more

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; Apply the formula of term, i.e., to find the ... more

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; Apply the formula of term, i.e., to find the ... more

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; Apply the formula of term, i.e., to find the ... more

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; Apply the formula of term, i.e., to find the ... more

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; Apply the formula of term, i.e., to find the ... more

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; Apply the formula of term, i.e., to find the ... more

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; Apply the formula of term, i.e., to find the ... more

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; Apply the formula of term, i.e., to find the ... more

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; Substitute and in the formula to obtain the ... more

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. ; Substitute and in the formula to obtain the... more

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; Substitute and in the formula to obtain the ... more

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. ; Substitute and in the formula to obtain the... more

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; Substitute and in the formula to obtain the ... more

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; Substitute and in the formula to obtain the ... more

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; Substitute and in the formula to obtain the ... more

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; Substitute and in the formula to obtain the ... more

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; Substitute and in the formula to obtain the ... more

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; Substitute and in the formula to obtain the ... more

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; The sequence is geometric in nature with the ... more

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; The general term in a geometric series is ... more

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; The general term in a geometric series is ... more

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; The general term in a geometric series is ... more

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; The general term in a geometric series is ... more

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; The general term in a geometric series is ... more

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13122 ; Any term in a geometric series can be ... more

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20480 ; Any term in a geometric series can be ... more

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; Any term in a geometric series can be written ... more

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; Any term in a geometric series can be written ... more

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9 ; The common ratio of a geometric sequence can ... more

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12 ; The common ratio of a geometric sequence can ... more

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-2 ; The common ratio of a geometric sequence can ... more

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; The common ratio of a geometric sequence can be... more

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The first three terms in the sequence are 18, 12, ... more

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The first 10 values of the sequence given by the ... more

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The first 10 values of the sequence given by the ... more

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The first 10 values of the sequence given by the ... more

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5461 ; Expand the finite sum to obtain the first ... more

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113.33 ; The finite sum is expanded, and the first... more

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-14706 ; Comparing with provides the first term... more

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; Comparing with provides the first term , the ... more

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; The given series can be written as, Comparing ... more

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12.5 ; The given series can be written as  ... more

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; The given function can be written as  Comparing... more

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; The given function can be written as   ... more

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1.6 ; The given function can be written as  ... more

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45 ; The given function can be written as . ... more

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; By comparing the sequence with the standard ... more

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; By comparing the sequence with the standard ... more

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; By comparing the sequence with the standard ... more

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. ; The number of terms in the sequence is 5, ... more

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2 ; Comparing the standard form with results the... more

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8 ; Comparing the standard form with results the... more

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; Compare with results the first term and the ... more

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; Compare with results in the first term and ... more

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5 ; Compare with results the first term and the... more

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5 ; Compare with results with the first term ... more

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32. ; By comparing the sequence with the standard ... more

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27 ; By comparing the sequence with the standard ... more

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No sum. ; By comparing the sequence with the ... more

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No sum ; By comparing the sequence with the ... more

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; Express the repeating decimal in an infinite ... more

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; Express the repeating decimal as an infinite ... more

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The curve is asymptotic to the line and the sum of... more

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The curve is asymptotic to the line and the sum of... more

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; The machine is worth and it depreciates at the... more

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Compare the sequence with the standard sum of ... more

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Approximately ; Let the sum of the infinite ... more

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; The tax rebate is and the rate is which can ... more

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; The tax rebate is and the rate is which can ... more

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approximately ; Considering the side of the ... more

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; Initially, the ball was dropped from a height... more

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; The salary for the first year is , and there is... more

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False. The ratio of consecutive terms should be ... more

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False. The term is the common ratio to the power... more

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The next term in the geometric sequence can be ... more

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Proof:Prove the formula for .Assume that the ... more

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 Proof:Prove the formula for .Assume that the ... more

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Proof:Assume that the inequality is true for .... more

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Proof:Assume that the formula is true for .Prove ... more

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  To prove its validity, mathematical induction is... more

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; The quadratic model has the form . Substitute ... more

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Mathematical induction. ; The first step in ... more

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First. ; First differences are the difference ... more

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Arithmetic. ; Arithmetic sequences are defined by ... more

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Second. ; A perfect quadratic model is when the ... more

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; Determine the statement for the given by ... more

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; Determine the statement for the given by ... more

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; Determine the statement for the given by ... more

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; Determine the statement for the given by ... more

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; Determine the statement for the given by ... more

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; Determine the statement for the given by ... more

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Rewrite the given expressionWhen , the statement ... more

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Rewrite the given expression    When , the ... more

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Rewrite the given expression   When , the ... more

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Rewrite the given expression   When , the ... more

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Rewrite the given expression When , the statement ... more

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Rewrite the given expression When , the statement ... more

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Rewrite the given expressionWhen , the statement ... more

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Rewrite the given expression When , the statement ... more

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Rewrite the given expression When , the statement ... more

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Rewrite the given expression When , the statement ... more

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Consider the given expression.  Show the above ... more

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Consider the given statement.For , show that the ... more

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Consider the given statement. For , show the above... more

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The mathematical expression is same as  The ... more

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Consider the inequality as a statement. Show the ... more

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Consider inequality as a statement.Show this ... more

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Consider inequality as a statement. Show the ... more

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Consider the inequality as a statement. Show the ... more

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Consider inequality as a statement. Show the ... more

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Consider the inequality as a statement. Show the ... more

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Consider a statement : is divisible by 3 for all... more

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Consider a statement : is divisible by 2 for all... more

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Consider a statement : is divisible by 3 for all... more

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Consider a statement : is divisible by 5 for all... more

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Consider a statement : for all  . Show the ... more

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Consider a statement : for all  .Show the ... more

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Consider a statement : ,  when  for all  . Show ... more

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Consider a statement : ,  when  for all  . Show ... more

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Consider a statement : for all  . Show the ... more

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Consider a statement :'' and are complex ... more

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 Proof:The number in an arithmetic sequence is ... more

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Proof:The number in a geometric sequence is given... more

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. Proof: Write first few sums of the given series ... more

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 Proof:Write the first few sums of the given ... more

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120 ; The sum of the first positive integer is . ... more

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465 ; The sum of the first positive integer is . ... more

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91 ; The sum of squares of first integers is . ... more

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3025. ; The sum of cubes of first integers is  . ... more

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979 ; The sum of fourth power of first integers ... more

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61776 ; The sum of fifth power of first integers ... more

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70 ; Distribute the summation over the addition in... more

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43890 ; Distribute the summation over the addition... more

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-3402 ; Distribute the summation over the addition... more

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195 ; Distribute the summation over the addition ... more

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Linear model; ; Find the first difference of the ... more

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Linear model; ; Find the first difference of the ... more

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Quadratic model; ; Find the first difference of ... more

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Quadratic model; ; Find the first difference of ... more

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Quadratic model; ; Find the first difference of ... more

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Quadratic model; ; Find the first difference of ... more

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The first difference is 3.The second difference is... more

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. ; Let the quadratic model be in the form  . ... more

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; Let the quadratic model be in the form . ... more

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; Let the quadratic model be in the form . ... more

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; Let the quadratic model be in the form . ... more

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. ; Let the quadratic model be in the form . ... more

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; Let the quadratic model be in the form . ... more

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; Let the linear model be . Substitute in to ... more

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False; must be proven to be true. ; If the series ... more

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Consider an example of a sequence with terms.The ... more

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; Take the eighth row of Pascal's triangle. The ... more

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; In the fourth row of Pascal's triangle, the ... more

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; In the third row of Pascal's triangle, the ... more

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Expanding. ; The binomial is solved by expanding ... more

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Binomial coefficient. ; The binomial is solved by ... more

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Binomial theorem. ; The binomial is solved by ... more

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; The binomial is solved by expanding the ... more

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10 ; Using the definition of a binomial ... more

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7 ; Using the definition of a binomial coefficient... more

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1 ; Using the definition of a binomial coefficient... more

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1 ; Using the definition of a binomial coefficient... more

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210 ; Using the definition of a binomial ... more

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210 ; The binomial coefficient can be evaluated ... more

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4950 ; The binomial coefficient can be evaluated ... more

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4950 ; The binomial coefficient can be evaluated ... more

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20 ; Binomial coefficients can be obtained by ... more

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6 ; Binomial coefficients can be obtained by ... more

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5 ; Binomial coefficients can be obtained by ... more

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35 ; Binomial coefficients can be obtained by ... more

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; The binomial theorem helps expand the ... more

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; The binomial theorem helps expand the ... more

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; The binomial theorem helps expand the ... more

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; The binomial theorem helps expand the ... more

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; The binomial theorem helps expand the ... more

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; The binomial theorem helps expand the ... more

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; The binomial theorem helps expand the ... more

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; The binomial theorem helps expand the ... more

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; The expression is expanded using the binomial ... more

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; The expression is expanded using the binomial ... more

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; The expression is expanded using the binomial ... more

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; The expression is expanded using the binomial ... more

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; The expression is expanded using the binomial ... more

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; To expand an expression using Pascal's ... more

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. ; To expand an expression using Pascal's ... more

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