Chapter 11.1, Problem 64HP

To determine

Find next term of the sequence $8,6,\frac{9}{2},\frac{27}{8},\mathrm{.....}$

Answer to Problem 64HP

Option D is correct.

Explanation of Solution

Given:

The sequence:$8,6,\frac{9}{2},\frac{27}{8},\mathrm{.....}$

Concept Used:

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

For example, the sequence $2,6,18,54,\mathrm{...}$ is a geometric progression with common ratio 3.

The ration of 2^nd term and first term is called the common ratio.

Calculation:

The sequence:$8,6,\frac{9}{2},\frac{27}{8},\mathrm{.....}$

Each term in a geometric sequence can be expressed in terms of the first term $a_1$ and the common ratio $r$ . Since each succeeding term is formulated from one or more previous terms, this is a recursive formula. First term $a_1=8$ and the common difference:$r=\frac{\text{2nd}\text{term}}{\text{1st}\text{term}}=\frac{6}{8}=\frac{3}{4}$

Terms	Symbol	In terms of $a_1$and r	Numbers
Fifth term	$a_5$	$a_1·r^4$	$8·{(\frac{3}{4})}^4=8·(\frac{81}{256})=\frac{81}{32}$

The next term of the sequence $8,6,\frac{9}{2},\frac{27}{8},\mathrm{.....}$ is $\frac{81}{32}$

Thus, the next term is: $\frac{81}{32}$ ; Option D is correct.