Chapter 11.3, Problem 69HP

To determine

Find the eighth term of the series.

Answer to Problem 69HP

  $a_8=234$

Explanation of Solution

Given:

The fifth term of a geometric sequence is $\frac{1}{27}$ th of the eighth term. If the ninth term is 702, What is the eighth term?

Concept Used:

Given: $a_5=\frac{a_8}{27}\text{and}{a}_9=702$

Calculation:

Given: $a_5=\frac{a_8}{27}\text{and}{a}_9=702$

We know that: $a_8=a_1·r^{8-1}=a_1·r^7\text{and}{a}_5=a_1·r^{5-1}=a_1·r^4$

Step 1: Find the value of r :

  $\begin{array}{l}{a}_5=\frac{a_8}{27}\\ 27·a_5=27·\frac{a_8}{27}\\ a_8=27·a_5\\ a_1·r^{8-1}=27·a_1·r^{5-1}\\ a_1·r^7=27·a_1·r^4\\ \frac{a_1·r^7}{a_1·r^4}=\frac{27·a_1·r^4}{a_1·r^4}\\ r^{7-4}=27[\frac{a^{ m}}{a^n}=a^{m-n}]\\ r^3=27\Rightarrow r=3\end{array}$

  $\begin{array}{l}{a}_9=a_8·r\\ 702=a_8·3\\ \frac{702}{3}=\frac{a_8·3}{3}\\ a_8=234\end{array}$

Thus, $a_8=234$