Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P²Rⁿ = Sⁿ.

Let the G.P. be a, ar, ar², ar³, …, arⁿ⁻¹.

According to the given information,

S = a(rⁿ − 1) / (r − 1)

P = aⁿ × r^(1 + 2 + … + (n − 1))

= aⁿ × r^[n(n − 1)/2]     [Sum of first n natural numbers = n(n − 1)/2]

R = 1/a + 1/(ar) + … + 1/(arⁿ⁻¹)

= (rⁿ⁻¹ + rⁿ⁻² + … + r + 1) / (a rⁿ⁻¹)

= (rⁿ − 1) / [a rⁿ⁻¹ (r − 1)]     [Since 1, r, r², …, rⁿ⁻¹ forms a G.P.]

∴ P²Rⁿ = a²ⁿ rⁿ(n − 1) × [(rⁿ − 1) / (a rⁿ⁻¹ (r − 1))]ⁿ

= [aⁿ (rⁿ − 1)ⁿ] / (r − 1)ⁿ

= [a(rⁿ − 1) / (r − 1)]ⁿ

= Sⁿ

Hence, P²Rⁿ = Sⁿ.