Question

Find the next three terms in each geometric sequence.

$2,-6,18,\dots$

Short answer

The next three terms in the geometric sequence is $-54,162,-486$.

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, thesequence2, 6, 18, 54, ... is ageometric progressionwith common ratio 3.

The ration of 2^nd term and first term is called the common ratio.

Step 2. Find the common difference. 

The sequence:$2,-6,18\mathrm{.....}$

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 $a_1=2$and the common difference is:

$$\begin{gathered} r=\frac{a_2}{a_1} \\ =\frac{-6}{2} \\ =-3 \end{gathered}$$

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 2 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	$a_4$	$a_1\cdot r^3$	$2{(-3)}^3=2\cdot \left(-27\right)=-54$
Fifth term	$a_5$	$a_1\cdot r^4$	$2\times {(-3)}^4=2\times 81=162$
Sixth term	$a_6$	$a_1\cdot r^5$	$2\times {(-3)}^5=2\times \left(-243\right)=-486$

Thus, next three terms of the sequence$2,-6,18\mathrm{.....}$are$-54,162,-486$.