Chapter 8.3, Problem 121E

To determine

To Explain: To find the nth term of a geometric sequence from the first two terms with explanation.

Explanation of Solution

Given information: Given first two terms of the geometric sequence.

Let the first two terms of the geometric sequence are $a\text{and}ar$ with common ratio r.

So, the third term of the sequence will be

  $a_3=a\times r\times r=a{r}^2=a{r}^{3-2}$

Similarly, fourth term will be

  $a_4=a\times r\times r\times r=a{r}^3=a{r}^{4-1}$

Now, by consider the pattern

  $\begin{array}{l}{a}_1=a{r}^{1-1}=a\\ a_2=a{r}^{2-1}=ar\\ a_3=a{r}^{3-1}=a{r}^2\\ a_4=a{r}^{4-1}=a{r}^3\end{array}$

It can be observed from the above pattern that the nth term will be

  $a_n=a{r}^{n-1}$

Hence the nth term of the geometric sequence is $a_n=a{r}^{n-1}$