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Q52.

Expert-verifiedFound in: Page 596

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.**

** $\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{1}\mathbf{,}\mathbf{.}\mathbf{....}$**

The next three terms in the geometric sequence are $\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{\text{\hspace{0.17em}\hspace{0.17em}}}\mathbf{1}\mathbf{,}\mathbf{\text{\hspace{0.17em}\hspace{0.17em}}}\mathbf{-}\mathbf{1}$.

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.

The sequence**: $-1,1,-1,1,\mathrm{.....}$**

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

**$\begin{array}{c}r=\text{\hspace{0.17em}\hspace{0.17em}}\frac{{a}_{2}}{{a}_{1}}\\ =\frac{1}{(-1)}\\ =-1\end{array}$**

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 | ${a}_{4}$ | ${a}_{1}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}{r}^{3}$ | $(-1){(-1)}^{3}=(-1)\times (-1)=1$ |

Fifth term | ${a}_{5}$ | ${a}_{1}\cdot {r}^{4}$ | $(-1){(-1)}^{4}=(-1)\times \left(1\right)=-1$ |

Sixth term | ${a}_{6}$ | ${a}_{1}\cdot {r}^{5}$ | $(-1){(-1)}^{5}=(-1)\times (-1)=1$ |

** **

Thus, next three terms of the sequence $-1,1,-1,1,\mathrm{.....}$ are** $-1,\text{\hspace{0.17em}\hspace{0.17em}}1,\text{\hspace{0.17em}\hspace{0.17em}}-1$**.

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