Q52. - Algebra 1

Question

Find the next three terms in each geometric sequence.

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

Step-by-Step Solution

Answer

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

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:  $-1,1,-1,1,\mathrm{.....}$

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:

$\begin{array}{c}r=\frac{a_2}{a_1}\\ =\frac{1}{(-1)}\\ =-1\end{array}$

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 $-1$ 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	$a_5$	$a_1\cdot r^4$	$(-1){(-1)}^4=(-1)\times \left(1\right)=-1$
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 $-1,1,-1$.