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Q52.

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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.
$- 1, 1, - 1, 1, . ....$

The next three terms in the geometric sequence are $- 1, 1, - 1$ .

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: $- 1, 1, - 1, 1, .....$

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 $a_{1} = - 1$ and the common ratio is:

$\begin{matrix} r = \frac{a_{2}}{a_{1}} \\ = \frac{1}{(- 1)} \\ = - 1 \end{matrix}$

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 $- 1$ 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	$a_{4}$	$a_{1} \cdot r^{3}$	$(- 1) {(- 1)}^{3} = (- 1) \times (- 1) = 1$
Fifth term	$a_{5}$	$a_{1} \cdot r^{4}$	$(- 1) {(- 1)}^{4} = (- 1) \times (1) = - 1$
Sixth term	$a_{6}$	$a_{1} \cdot r^{5}$	$(- 1) {(- 1)}^{5} = (- 1) \times (- 1) = 1$

Thus, next three terms of the sequence $- 1, 1, - 1, 1, .....$ are $- 1, 1, - 1$ .

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