The 5th, 8th and 11th terms of a G.P. are p, q and s respectively. Show that q² = ps.

Let the first term of the G.P. be a and common ratio = r.

Given,

⇒ a₅ = ar⁴ = p

⇒ a₈ = ar⁷ = q

⇒ a₁₁ = ar¹⁰ = s

We need to prove q² = ps.

L.H.S. = q²

⇒ q² = q × q = ar⁷ × ar⁷ = a²r¹⁴.

R.H.S. = ps

⇒ ps = p × s = ar⁴ × ar¹⁰ = a²r¹⁴.

∴ L.H.S. = R.H.S. = a²r¹⁴.

Hence, proved that q² = ps.