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Chapter 12: Q. 58 (page 817)

How many terms must be added in an arithmetic sequence whose first term is 78 and whose common difference is -4 to obtain a sum of 702?

Short Answer

Expert verified

Number of terms either 27 or 13

Step by step solution

01

Given information

In an arithmetic sequence,

first term is 78 , a1=78

common difference is -4, d=-4

sum is 702

02

Sum formula

$S_{n} = \frac{n}{2} (2 a_{1} + (n - 1) d) 702 = \frac{n}{2} (2 (78) + (n - 1) (- 4)) m u l t i p l y b y 2 o n b o t h s i d e s 1404 = n (156 - 4 n + 4) 1404 = - 4 n^{2} + 160 n 4 n^{2} - 160 n + 1404 = 0$

03

Quadratic formula

Solve for n using quadratic formula

$4 n^{2} - 160 n + 1404 = 0 n = \frac{- (- 160) \pm \sqrt{{(- 160)}^{2} - 4 \cdot 4 \cdot 1404}}{2 \cdot 4} n = \frac{- (- 160) \pm 56}{8} n = \frac{- (- 160) + 56}{8} = 27 n = \frac{- (- 160) - 56}{8} = 13$

Number of terms either 27 or 13

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Most popular questions from this chapter

In Problems 17–28, write down the first five terms of each sequence.

$\{a_{n}\} = \{\frac{3^{n}}{n}\}$

In Problems 71-82, find the sum of each sequence.

${\sum (-}_{k = 1}^{24} k)$

In Problems 51–66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.

$8 + 4 + 2 + . . . .$

In Problems 61–70, express each sum using summation notation.

$\frac{2}{3} - \frac{4}{9} + \frac{8}{27} - . . . . . . . . + {(- 1)}^{12} {(\frac{2}{3})}^{11}$

In Problems 51–60, write out each sum.

$\sum_{k = 2}^{n} {(- 1)}^{k} In k$

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