Question

If 2, a, b, −54 forms a geometric sequence, find the values of a and b.

Answer 1

Answer:

The values are a=-6 and b=18.

Step-by-step explanation:

A Geometric sequence is given by the formula   $a_n=mr^{n-1}$ for all $n\geq 1$, where r ≠ 0 is the common ratio and $m$ is the first term of the sequence .

In this problem we know that m= 2, $a=a_{2}$ is the second term of the sequence and  $a_{3} =b$ is the third term.

First we need to find the general form of the sequence, we can use the fourth term $a_{4}=-54$ and the value of $m=2$ to find r.

We replace  $a_{4}=-54$ and  $m=2$ in the formula  $a_n=mr^{n-1}$, then we have

$a_4=mr^{4-1}$

$-54=2r^{4-1}$

$\frac{-54}{2} =r^3$

$-27=r^3$

$r=-3$.

Therefore the general form of the sequence is  $a_n=2(-3)^{n-1}$ for all $n \geq 1$.

To find the value of a, we replace n=2 in our formula, so

$a=a_2=2(-3)^{2-1}=2(-3)=-6$.

To find the value of b, we replace n=3 in our formula, so

$b=a_3=2(-3)^{3-1}=2(-3)^{2} =2(9)=18$.