Question

In Problems 7–12, determine whether the given sequence 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 the sum of the first n terms.

$\left\{s_n\right\}=\left\{2^{3n}\right\}$

Short answer

• Given sequence is a geometric sequence.
• The common ratio of the sequence is 8.
• Th sum of first $n$terms is:$S_n=\frac{8}{7}\left(8^n-1\right)$

Step-by-step solution

Step 1. Given information

Sequence:$\left\{s_n\right\}=\left\{2^{3n}\right\}$

Step 2. To find first term and n-1th term.

$$\begin{gathered} s_1=2^{3.1} \\ =2^3 \\ =8 \end{gathered}$$

$$\begin{gathered} s_{n-1}=2^{3(n-1)} \\ =2^{3n-3} \\ =\frac{2^{3n}}{2^3} \end{gathered}$$

Step 3. To find ratio between two successive terms.

Ratio between two successive terms is:

$$\begin{gathered} r=\frac{s_n}{s_{n-1}} \\ =\frac{2^{3n}}{{\displaystyle \frac{2^{3n}}{2^3}}} \\ =2^3 \\ =8 \end{gathered}$$

Since ratio is constant.

Hence the given sequence is a geometric sequence whose common ratio is 8.

Step 4. To find sum of first n term of the sequence.

$$\begin{gathered} S_n=\frac{8\left(8^n-1\right)}{8-1} \\ {S}_n=\frac{8}{7}\left(8^n-1\right) \end{gathered}$$