Chapter 9, Problem 54SGR

To determine

Find next three terms of the sequence.

Answer to Problem 54SGR

The next three terms are $32,16,8,$

Explanation of Solution

Given:

The sequence: $256,128,64,\mathrm{.....}$

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: $256,128,64,\mathrm{.....}$

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 $a_1=256$ and the common difference: $r=\frac{a_2}{a_1}=\frac{128}{256}=\frac{1}{2}$

Terms	Symbol	In terms of $a_1$ and r	Numbers
Fourth	$a_4$	$a_1·r^3$	$256·{(\frac{1}{2})}^3=256·\frac{1}{8}=32$
Fifth term	$a_5$	$a_1·r^4$	$256·{(\frac{1}{2})}^4=256·\frac{1}{16}=16$
Sixth term	$a_6$	$a_1·r^5$	$256·{(\frac{1}{2})}^5=256·\frac{1}{32}=8$

The next three terms of the sequence $256,128,64,\mathrm{.....}$ are $32,16,8,$

Thus, the next three terms are: $32,16,8,$