The formula 1 + r + r 2 ++ r n = r n+1 1 r 1 Cengage Learning. All Rights Reserved. May

Chapter 5, Problem 2

(choose chapter or problem)

The formula

\(1+r+r^{2}+\cdots+r^{n}=\frac{r^{n+1}-1}{r-1}\)

is true for all real numbers r except r = 1 and for all integers n ≥ 0. Use this fact to solve each of the following problems:

a. If i is an integer and i ≥ 1, find a formula for the expression \(1 + 2 + 2^{2} +· · ·+2^{i−1}\).

b. If n is an integer and n ≥ 1, find a formula for the expression \(3^{n−1} + 3^{n−2} +· · ·+3^{2} + 3 + 1\).

c. If n is an integer and n ≥ 2, find a formula for the expression \(2^{n} + 2^{n−2} ·3 + 2^{n−3} ·3+· · ·+2^{2} ·3 + 2·3 + 3\)

d. If n is an integer and n ≥ 1, find a formula for the expression

\(2^{n}-2^{n-1}+2^{n-2}-2^{n-3}+\cdots+(-1)^{n-1} \cdot 2+(-1)^{n}\).

Text Transcription:

1+r+r^2+cdots+r^n= r^n+1-1 / r-1

1 + 2 + 2^2 +· · ·+2^i−1

3^n−1 + 3^n−2 +· · ·+3^2 + 3 + 1

2^n + 2^n−2 ·3 + 2^n−3 ·3+· · ·+2^2 ·3 + 2·3 + 3

2^n-  2^n-1 + 2^n-2 -  2^n-3 + cdots + (-1)^n-1 cdot 2 + (-1)^n