Question

If 24, x, and 6 are in GP, then find the value of x.

Answer 1

Hint: Geometric progression is a progression of numbers with a constant ratio between each number and the previous number. The first three terms of the GP are \[a,ar,a{r^2}\]. Use this to find the value of x.

Complete step-by-step answer:
Geometric progression (GP) is a sequence of numbers, each of which differs from the succeeding one by a constant ratio. The terms of a geometric progression are obtained by multiplying the constant ratio with the number to get the next number and so on.
The first term of a GP is represented by the letter ‘a’ and the constant ratio is also called the common ratio and is represented by the letter ‘r’.
We are given three numbers which are in GP as below:
24, x, 6
We observe that 24 is the first term, hence, we have:
\[a = 24............(1)\]
Let the common ratio be r, then the second term is given as \[ar\]. Hence, we have:
\[x = ar..............(2)\]
The third term of a GP is given by \[a{r^2}\] and is equal to the number 6, hence, we have:
\[a{r^2} = 6.............(3)\]
Substituting equation (1) in equation (3), we get:
 \[(24){r^2} = 6\]
Solving for r, we get:
\[{r^2} = \dfrac{6}{{24}}\]
\[{r^2} = \dfrac{1}{4}\]
\[r = \pm \dfrac{1}{2}..............(4)\]
Substituting equation (1) and equation (4) in equation (2), we get:
\[x = 24 \times \left( { \pm \dfrac{1}{2}} \right)\]
\[x = \pm 12\]
Hence, the value of x is 12 or – 12.

Note: You have two solutions for x and you need to write both of them, otherwise your answer won’t be given full credit. You can also use the formula \[ac = {b^2}\] to find the value of x, where a, b and c are consecutive terms of any GP.