Chapter 9, Problem 23PT

To determine

Find next three terms of the sequence.

Answer to Problem 23PT

The next three terms are $125,62.5,31.25,$

Explanation of Solution

Given:

The sequence: $1000,500,250\mathrm{.....}$

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: $1000,500,250\mathrm{.....}$

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 $a_1=1000$ and the common difference: $r=\frac{a_2}{a_1}=\frac{500}{1000}=\frac{1}{2}$

Terms	Symbol	In terms of $a_1$ and r	Numbers
Fourth term	$a_4$	$a_1·r^3$	$1000·{(\frac{1}{2})}^3=1000·(\frac{1}{8})=125$
Fifth term	$a_5$	$a_1·r^4$	$1000·{(\frac{1}{2})}^4=1000·(\frac{1}{16})=62.5$
Sixth term	$a_6$	$a_1·r^5$	$1000·{(\frac{1}{2})}^5=1000·(\frac{1}{32})=31.25$

The next three terms of the sequence $1000,500,250\mathrm{.....}$ are $125,62.5,31.25,$

Thus, the next three terms are: $125,62.5,31.25,$