Chapter 12.3, Problem 9AYU

In problems $9-18$, show that each sequence is geometric. Then find the common ratio and list the first four terms.

$\left\{s_n\right\}=\left\{4^n\right\}$

To determine

The common ratio and first four terms of the sequence $\left\{s_n\right\}=\left\{4^n\right\}$ by showing the sequence is a geometric sequence.

Answer to Problem 9AYU

Solution:

The common ratio is $4$.

The first four terms are $4,16,64,256$.

Explanation of Solution

Given information:

The sequence $\left\{s_n\right\}=\left\{4^n\right\}$

Explanation:

The given sequence is $\left\{s_n\right\}=\left\{4^n\right\}$

The first term of the sequence is $s_1=4^1=4$

The $n^{th}$ term and ${\left(n-1\right)}^{th}$ term of the sequence are $s_n=4^n$ and $s_{n-1}=4^{\left(n-1\right)}$ respectively.

The common ratio is given by $r=\frac{s_n}{s_{n-1}}$.

The ratio of consecutive terms is $\frac{s_n}{s_{n-1}}=\frac{4^n}{4^{\left(n-1\right)}}$

$=4^n÷4^{\left(n-1\right)}=4^{\left(n-n+1\right)}$

$=4$

As the common ratio of consecutive terms is constant, the sequence $\left\{s_n\right\}=\left\{4^n\right\}$ is a geometric sequence with common ratio $4$.

Now, to calculate first four terms, substitute $n=1,2,3,4$ in $\left\{s_n\right\}=\left\{4^n\right\}$

$s_1=4^1=4$.

$s_2=4^2=16$.

$s_3=4^3=64$.

$s_4=4^4=256$.

Therefore, the first four terms of the given sequence are $4,16,64,256$.