Chapter 12, Problem 7CT

To determine

Whether the sequence $\left\{-\frac{1}{2}\cdot 4^n\right\}$ is arithmetic, geometric, or neither. If the sequence is arithmetic find the common difference and the sum of the first n terms. If the sequence is geometric find the common ratio and sum of the first n terms.

Answer to Problem 7CT

Solution:

The sequence $\left\{-\frac{1}{2}\cdot 4^n\right\}$ is geometric with common ratio $r=4$ and sum of first n terms is

$S_n=\frac{2}{3}\left(1-4^n\right)$.

Explanation of Solution

Given information:

Sequence, $\left\{-\frac{1}{2}\cdot 4^n\right\}$.

Explanation:

Consider the sequence, $\left\{-\frac{1}{2}\cdot 4^n\right\}$. n^th term of the sequence is $a_n=-\frac{1}{2}\cdot 4^n$ and (n-1)^th term is $a_{n-1}=-\frac{1}{2}\cdot 4^{n-1}$

Let’s check if the ratio of consecutive terms is constant.

$\frac{a_n}{a_{n-1}}=\frac{-\frac{1}{2}\cdot 4^n}{-\frac{1}{2}\cdot 4^{n-1}}=\frac{-\frac{1}{2}\cdot 4^{n-1}\cdot 4}{-\frac{1}{2}\cdot 4^{n-1}}=4$

The ratio of consecutive terms is constant; hence the sequence is geometric with common ratio $r=4$.

The first term of the sequence is $a_1=-\frac{1}{2}\cdot 4^1=-2$.

The sum of the first n terms of the geometric sequence is given by

$S_n=a_1\cdot \frac{1-r^n}{1-r}$

$=-2\cdot \frac{1-4^n}{1-4}$

$=\frac{2}{3}\left(1-4^n\right)$

Thus the sequence $\left\{-\frac{1}{2}\cdot 4^n\right\}$ is geometric with common ratio $r=4$ and sum of first n terms is

$S_n=\frac{2}{3}\left(1-4^n\right)$.