Chapter 11.1, Problem 40PPS

To determine

Find next three terms of the sequence $18,12,8,\mathrm{.......}$

Answer to Problem 40PPS

The next three terms are $\frac{16}{3},\frac{32}{9},\frac{64}{27},$

Explanation of Solution

Given:

The sequence:$18,12,8,\mathrm{.......}$

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, ... 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:$18,12,8,\mathrm{.......}$

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 $a_1=18;$ and the common difference:$r=\frac{\text{2nd}\text{term}}{\text{1st}\text{term}}=\frac{12}{18}=\frac{2}{3}$

Terms	Symbol	In terms of $a_1$and r	Numbers
Fourth term	$a_4$	$a_1·r^3$	$18·{(\frac{2}{3})}^3=18·(\frac{8}{27})=\frac{16}{3}$
Fifth term	$a_5$	$a_1·r^4$	$18·{(\frac{2}{3})}^4=18·(\frac{16}{81})=\frac{32}{9}$
Sixth term	$a_6$	$a_1·r^5$	$18·{(\frac{2}{3})}^5=18·(\frac{32}{243})=\frac{64}{27}$

The next three terms of the sequence $18,12,8,\mathrm{.......}$ are $\frac{16}{3},\frac{32}{9},\frac{64}{27},$

Graph the terms:

  

Thus, the next three terms are: $\frac{16}{3},\frac{32}{9},\frac{64}{27},$