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Q54.

Expert-verified
Found in: Page 596

### Algebra 1

Book edition Student Edition
Author(s) Carter, Cuevas, Day, Holiday, Luchin
Pages 801 pages
ISBN 9780078884801

# Find the next three terms in each geometric sequence.$\mathbf{256}\mathbf{,}\mathbf{128}\mathbf{,}\mathbf{64}\mathbf{,}\mathbf{.}\mathbf{....}$

The next three terms in the geometric sequence are $\mathbf{32}\mathbf{,}\mathbf{\text{\hspace{0.17em}\hspace{0.17em}}}\mathbf{16}\mathbf{,}\mathbf{\text{\hspace{0.17em}\hspace{0.17em}}}\mathbf{8}$.

See the step by step solution

## 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 ${2}^{nd}$ 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:

$\begin{array}{c}r=\text{\hspace{0.17em}\hspace{0.17em}}\frac{{a}_{2}}{{a}_{1}}\\ =\frac{128}{256}\\ =\frac{1}{2}\end{array}$

## 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 ${a}_{4}$ ${a}_{1}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}{r}^{3}$ Fifth term ${a}_{5}$ ${a}_{1}\cdot {r}^{4}$ Sixth term ${a}_{6}$ ${a}_{1}\cdot {r}^{5}$

Thus, next three terms of the sequence $256,128,64,.....$ arelocalid="1647772115548" $32,\text{\hspace{0.17em}\hspace{0.17em}}16,\text{\hspace{0.17em}\hspace{0.17em}}8$.