Question

Find the common ratio & the next three terms in the sequence.

Answer 1

Answer:

We conclude that:

• Common ratio: r = -3

• Next three terms: 162, -486, 1458

Hence, option D is correct.

Step-by-step explanation:

Given the geometric sequence

2, -6, 18, -54, ...

Here:

$a_1=2$

A geometric sequence has a constant difference 'r' and is defined by  

$a_n=a_1\cdot \:r^{n-1}$

computing the differences of all the adjacent terms

$\frac{-6}{2}=-3,\:\quad \frac{18}{-6}=-3,\:\quad \frac{-54}{18}=-3$

The ratio of all the adjacent terms is the same and equal to

$r=-3$

now substituting $a_1=2$ and $r=-3$ in the nth term of the sequence

$a_n=a_1\cdot \:r^{n-1}$

now substituting n = 5 to determine the 5th term

$a_5=2\left(-3\right)^{5-1}$

$a_5=3^4\cdot \:2$

$a_5=2\cdot \:81$

$a_5=162$

now substituting n = 6 to determine the 6th term

$a_6=2\left(-3\right)^{6-1}$

$a_6=-2\cdot \:243$

$a_6=-486$

now substituting n = 7 to determine the 7th term

$a_7=2\left(-3\right)^{7-1}$

$a_7=2\cdot \:729$

$a_7=1458$

Therefore, we conclude that:

• Common ratio: r = -3

• Next three terms: 162, -486, 1458

Hence, option D is correct.