Given: 5^th, 8^th and 11^th terms of a G.P. are p, q and s, respectively

We know that in G.P a_n = ar^n-1

Here, n: number of terms

a: First term

r: common ratio

Here,

a₅ = ar^5-1 = ar⁴

⇒ p = ar⁴ (∵ 5^th term of G.P. is given p) –1

Similarly,

a₈ = ar^8-1 = ar⁷

⇒ q = ar⁷ (∵ 7^th term of G.P. is given q) –2

a₁₁ = ar^11-1 = ar¹⁰

⇒ s = ar¹⁰ (∵ 11^th term of G.P. is given s) –3

We can observe that:

q × q = p × s (that is, ar⁷ × ar⁷ = ar⁴ × ar¹⁰)

∴ q² = ps

Hence proved