Chapter 8.3, Problem 112E

To determine

Whether the given statement “quotient of 7^th to the 5^th term of a geometric sequence is 16, then common ratio will be 8” is true or false.

Answer to Problem 112E

Thegiven statement is false.

Explanation of Solution

In a sequence $a_1,a_2,a_3,a_4,\dots \dots ,a_n,\dots$ if ratio of the consecutive terms is the same then it is defined as Geometric Sequence.

So, a number r is defined as the common ratio of the geometric sequence if $\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}=\dots \dots =r.$

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.

Now, 5^th term and 7^th term of a geometric sequence will be $a_1{r}^4$ and $a_1{r}^6$ respectively.

So, the quotient of 7^th term to the 5^th term will be

  $\begin{array}{c}\frac{a_1{r}^6}{a_1{r}^4}=r^2\\ 16=r^2\\ r=4\end{array}$

So, the common ratio will be 4 not 8.

Hence, the given statement is false.