The sum of the first and second term in a gp is 108 and the sum of the third and fourth term is 12. Find the 2 possible values for the 1st 2 terms

Question

This is a sum from the chapter review section of my book and the answer is 81, 27 or 162, -54. But I didn't understand the process.

Answer 1

let first term be $a$ and common ratio $r$.
Then, series will proceed as: $a , ar, ar^2, ar^3$

$a + ar = 108$
$ar^2 + ar^3 = 12$

On simplifying the two equations,

$a (1+r)= 108$ ... (i)
$ar^2 (1+r)=12$ ...(ii)

On dividing the two equations, we get $r= +\frac{1}{3}$ or $-\frac{1}{3}$.

On substituting $r=+\frac{1}{3}$ back into equation (i), you can compute the value of $a$ to be $81$ and thus $ar$ (second term) will be equal to $27$.

On substituting the $r=-\frac{1}{3}$ into equation (ii), you can compute the value of $a$ to be $162$ and this $ar$ (second term) will be equal to $-54$.

Hope this helps.