Chapter 12, Problem 7T

To determine

To show: The sequence $a_1^2,a_2^2,a_3^2\cdots$ is a geometric sequence.

Explanation of Solution

Formula to calculate the nth term of the geometric sequence $a_n$ with first term $a$ and common ratio $r$ is,

$a_n=a{r}^{n-1}$ (1)

The given sequence $a_1,a_2,a_3\cdots$ is in geometric sequence with initial term $a$ and common ratio $r$.

Assume the value of $a_1$ is equal to a.

Substitute 2 for $n$ in equation (1), to evaluate $a_2$.

$\begin{array}{c}{a}_2=a{r}^{2-1}\\ =ar\end{array}$

Substitute 2 for $n$ in equation (1), to evaluate $a_3$,

$\begin{array}{c}{a}_3=a{r}^{3-1}\\ =a{r}^2\end{array}$

The given sequence can be written as $a,ar,a{r}^2,\cdots$.

Take squares of each term of the above sequence, then the new sequence is $a^2,a^2{r}^2,a^2{r}^4,\cdots$ .

First term of the new sequence is $a^2$, second term is $a^{2}r$ and third term is $a^4{r}^2$ and so on.

Divide the second term of the new sequence by first term,

$\frac{a^2{r}^2}{a^2}=r^2$.

Divide the third term of the new sequence by second term,

$\frac{a^2{r}^4}{a^2{r}^2}=r^2$

Since the common ratio is same that is $r^2$. The new sequence $a^2,a^2{r}^2,a^2{r}^4,\cdots$ is also geometric sequence with common ratio $r^2$ and first term $a^2$.

Hence, the sequence $a_1^2,a_2^2,a_3^2\cdots$ is in geometric sequence.