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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 12: Q 43. (page 825)

In Problems 39鈥� 44, find each sum.
$- 1 - 2 - 4 - 8 - . . . - (2^{n - 1})$

Short Answer

Expert verified

The sum of the given sequence is $1 - 2^{n}$

Step by step solution

01

Step 1. Write the given information.

The given sequence is $- 1 - 2 - 4 - 8 - . . . - (2^{n - 1})$

02

Step 2. Find the common ratio.

$a_{1} = - 1, a_{2} = - 2, a_{3} = - 4 \frac{a_{2}}{a_{1}} = \frac{- 2}{- 1} = 2 \frac{a_{3}}{a_{2}} = \frac{- 4}{- 2} = 2 r = 2$

03

Step 3. Find the sum.

The formula for the sum is $s_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}$

$s_{n} = - 1 路 \frac{1 - 2^{n}}{1 - 2} = 1 - 2^{n}$

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Most popular questions from this chapter

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

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

How much do you need to invest now at $5 %$ per annum compounded monthly so that in 1 year you will have $$ 10000$ ?

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

$\{c_{n}\} = \{\frac{n^{2}}{2^{n}}\}$

In Problems 29鈥�36, the given pattern continues. Write down the nth term of a sequence $\{a_{n}\}$ suggested by the pattern.

$1, \frac{1}{2}, 3, \frac{1}{4}, 5, \frac{1}{6}, 7, \frac{1}{8}, . . .$

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

{鈭�}_{k = 1}^{鈭�} 5 {(\frac{1}{4})}^{k - 1}

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