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Q54.

Expert-verified

Algebra 1

Found in: Page 596

Book edition Student Edition

Author(s) Carter, Cuevas, Day, Holiday, Luchin

Pages 801 pages

ISBN 9780078884801

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Short Answer

Find the next three terms in each geometric sequence.
$256, 128, 64, . ....$

The next three terms in the geometric sequence are $32, 16, 8$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. State the 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 $2^{n d}$ term and first term is called the common ratio.

Step 2. Find the common ratio.

The sequence: $256, 128, 64, .....$

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 ratio is:

$\begin{matrix} r = \frac{a_{2}}{a_{1}} \\ = \frac{128}{256} \\ = \frac{1}{2} \end{matrix}$

Step 3. Find the next three terms.

In order to calculate the next three terms of the geometric sequence substitute $n = 4, 5, 6$ and 256 for $a_{1}$ into the formula $a_{n} = a_{1} r^{n - 1}$ .

Terms	Symbol	In terms of $a_{1}$ and $r$	Numbers
Fourth term	$a_{4}$	$a_{1} \cdot r^{3}$	$256 {(\frac{1}{2})}^{3} = 256 \times \frac{1}{8} = 32$
Fifth term	$a_{5}$	$a_{1} \cdot r^{4}$	$256 {(\frac{1}{2})}^{4} = 256 \times \frac{1}{16} = 16$
Sixth term	$a_{6}$	$a_{1} \cdot r^{5}$	$256 {(\frac{1}{2})}^{5} = 256 \times \frac{1}{32} = 8$

Thus, next three terms of the sequence $256, 128, 64, .....$ arelocalid="1647772115548" $32, 16, 8$ .

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