The sum of the first 8 terms of a GP is five times the sum of its first 4 terms. If r 1 is the common ratio, then what is the number of possible real values of r ?

Step-by-step solution

Let the first term of the GP be a and the common ratio be r . The sum of the first n terms is given by S_n = a(r^n - 1) r - 1 . According to the given condition, S₈ = 5S₄ . a(r^8 - 1) r - 1 = 5 a(r^4 - 1) r - 1 For a non-zero first term a and given r 1 : r^8 - 1 = 5(r^4 - 1) (r^4 - 1)(r^4 + 1) = 5(r^4 - 1) Assuming r^4 - 1 0 to avoid the trivial zero-sum case: r^4 + 1 = 5 r^4 = 4 Since r is a real number, r^2 must be positive. r^2 = 2 r = 2 Therefore, there are two possible real values of r .