Chapter 11.1, Problem 39PPS

To determine

Find next three terms of the sequence $0.125,-0.5,2\mathrm{.....}$

Answer to Problem 39PPS

The next three terms are $-8,32,-124$

Explanation of Solution

Given:

The sequence:$0.125,-0.5,2\mathrm{.....}$

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:$0.125,-0.5,2\mathrm{.....}$

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 $a_1=0.125;$ and the common difference:$r=\frac{\text{2nd}\text{term}}{\text{1st}\text{term}}=\frac{-0.5}{0.125}=-4\text{or}r=\frac{\text{2}}{-\text{0}\text{.5}}=-4$

Terms	Symbol	In terms of $a_1$and r	Numbers
Fourth term	$a_4$	$a_1·r^3$	$0.125·{(-4)}^3=0.125·(-64)=-8$
Fifth term	$a_5$	$a_1·r^4$	$0.125·{(-4)}^4=0.125·(256)=32$
Sixth term	$a_6$	$a_1·r^5$	$0.125·{(-4)}^5=0.125·(-1024)=-124$

The next three terms of the sequence $0.125,-0.5,2\mathrm{.....}$ are $-8,32,-124$

The terms on the graph:

  

Thus, the next three terms are: $-8,32,-124$