The ratio between the sum of the first four terms of G.P. is to the sum of the first eight terms of G.P. is 1 : 17. Then, find the common ratio of the G.P?

Given:

S₄ : S₈ = 1 : 17

Formula used:

S_n = a(rⁿ – 1) /(r – 1)

Where, a = first term, r = common ratio and n = number of terms

S₄ = a(r⁴ – 1) /(r – 1)     ---- (1)

S₈ = a(r⁸ – 1) /(r – 1)     ---- (2)

Calculation:

S⁴: S⁸ = (r⁴ – 1) : (r⁸ – 1)

⇒ (r⁴ – 1)/(r⁸ – 1) = 1/17

⇒ (r⁴ – 1)/[ (r⁴ + 1) (r⁴ – 1) ] = 1/17

⇒ 1/(r⁴ + 1) = 1/17

⇒ (r⁴ + 1) = 17

⇒ r⁴ = 16

∴ r = 2