Chapter 11.3, Problem 68HP

To determine

Prove the formula for the nth term of a geometric sequence.

Answer to Problem 68HP

Explanation of Solution

Given:

Prove the formula for the nth term of a geometric sequence.

Concept Used:

Let $a_n$ is the nth term of the sequence and r = the common ratio.

Calculation:

Let $a_n$ is the nth term of the sequence and r = the common ratio.

Definition of the second term of a geometric sequence is $a_3=a_2·r$

Definition of the third term of a geometric sequence is $a_3=a_2·r$

Now substitute the value of $a_2=a_1·r$ in $a_3=a_2·r$

  $\begin{array}{l}{a}_3=a_2·r\\ a_3=a_1·r·r=a_1·r^2=a_1·r^{3-1}\end{array}$

Definition of the fourth term of a geometric sequence is $a_4=a_3·r$

Now substitute the value of $a_3=a_1·r^{3-1}$ in $a_4=a_3·r$

  $a_4=a_3·r=a_1·r^{3-1}=a_1·r^{4-1}$

  $a_4=a_1·r^{4-1}$

In the same way we can write the nth term of GP series: $a_n=a_1·r^{n-1}$

Thus, $a_n=a_1·r^{n-1}$