Chapter 12.3, Problem 8AYU

To determine

To fill: Check whether the given statement is true or false.

Answer to Problem 8AYU

Solution:

It is a true statement.

Explanation of Solution

Given:

It is a geometric sequence with first term $a_1$ and common ratio, where $r\ne 0,1$, te sum of the first $n$ terms is $S_n=a_1\cdot \frac{1-r^n}{1-r}$.

Calculation:

Let $\left\{a_n\right\}$ be a geometric sequence with first term $a_1$ and common ratio $r$, where $r\ne 0,r\ne 1$. The sum $S_n$ of the first $n$ terms of $\left\{a_n\right\}=\left\{a_1{r}^{n-1}\right\}$ is $S_n=a_1+a_{1}r+\cdot \cdot \cdot +a_1{r}^{n-1}$.   -----(1)

Multiply each side by $r$ obtain, $r{S}_n=a_{1}r+a_1{r}^2+\cdot \cdot \cdot +a_1{r}^n$.   -----(1)

Subtract (2) from (1). The result becomes $S_n-r{S}_n=a_1-a_1{r}^n$.

$S_n\left(1-r\right)=a_1\left(1-r^n\right)$

Since $r\ne 1,S_n=a_1\frac{\left(1-r^n\right)}{\left(1-r\right)}$.

Hence the given statement is true.