Chapter 11.1, Problem 41PPS

To determine

Find next three terms of the sequence $64,48,36,\mathrm{........}$

Answer to Problem 41PPS

The next three terms are $27,\frac{81}{4},\frac{243}{16},$

Explanation of Solution

Given:

The sequence:$64,48,36,\mathrm{........}$

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:$64,48,36,\mathrm{........}$

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 $a_1=64;$ and the common difference:$r=\frac{\text{2nd}\text{term}}{\text{1st}\text{term}}=\frac{48}{64}=\frac{3}{4}$

Terms	Symbol	In terms of $a_1$and r	Numbers
Fourth term	$a_4$	$a_1·r^3$	$64·{(\frac{3}{4})}^3=64·(\frac{27}{64})=27$
Fifth term	$a_5$	$a_1·r^4$	$64·{(\frac{3}{4})}^4=64·(\frac{81}{256})=\frac{81}{4}$
Sixth term	$a_6$	$a_1·r^5$	$64·{(\frac{3}{4})}^5=64·(\frac{243}{1024})=\frac{243}{16}$

The next three terms of the sequence $64,48,36,\mathrm{........}$ are $27,\frac{81}{4},\frac{243}{16},$

The terms on the graph:

  

Thus, the next three terms are: $27,\frac{81}{4},\frac{243}{16},$