Chapter 12.2, Problem 13AYU

In Problems 7-16, show that each sequence is arithmetic. Find the common difference and write out the first four terms.

$\left\{S_n\right\}=\left\{\ln 3^n\right\}$

To determine

To find: Each sequence is arithmetic and the common difference. Write out the first four terms $\left\{S_n\right\}=\left\{\ln 3^n\right\}$.

Answer to Problem 13AYU

$d=\ln 3$, $s_1=\ln 3$, $s_2=2\ln 3$, $s_3=3\ln 3$, $s_4=4\ln 3$

Explanation of Solution

Given:

$\left\{S_n\right\}=\left\{\ln 3^n\right\}$

Calculation:

$s_n=\ln 3^n=n\ln 3$

The $n^{\text{th}}$ and ${\left(n-1\right)}^{\text{th}}$ term of the sequence $\left\{s_n\right\}$ are

$s_n=n\ln 3 \text{and }{s}_{n-1}=n\ln 3-\ln 3$

$d=s_n-s_{n-1}=n\ln 3-n\ln 3+\ln 3=\ln 3$   (a constant)

Since the difference of any two successive term is the constant, the given sequence is arithmetic and the common difference is $\ln 3$.

The first four terms are,

$s_1=\ln 3$

$s_2=2\ln 3$

$s_3=3\ln 3$

$s_4=4\ln 3$