Chapter 8.3, Problem 114E

To determine

Whether the given statement “to find the $n^{th}$ term of a geometric sequence , multiply its common ratio by the first term of the sequence raised to the ${\left(n-1\right)}^{th}$ power ”is true or false.

Answer to Problem 114E

Thegiven statement is false.

Explanation of Solution

Given information :

Givenstatement is “to find the $n^{th}$ term of a geometric sequence , multiply its common ratio by the first term of the sequence raised to the ${\left(n-1\right)}^{th}$ power”

Since find the $n^{th}$ term of a geometric sequence is $a_1{r}^{\left(n-1\right)}$ , where $a_1$ is first term and $r$ is common ration of the geometric sequence.

For example: $1,3,9,27,\mathrm{.......}$ is a geometric sequence as its ratio of consecutive are same.

  $\begin{array}{l}i.e.\text{common}\text{ratio}\text{r}=\frac{3}{1}=\frac{9}{3}=\frac{27}{9}=3and\\ n^{th}termis{a}_n=a_1{\left(r\right)}^{n-1}\\ a_n=1{\left(3\right)}^{n-1}\\ \therefore a_n=3^{n-1}\\ \end{array}$

Hence, the given statement is false.