Integration with sequence and series...a very difficult and challenging question!!!Any maths expert please

If a n =limits from 0 to pi/2 integral of (sin^2nx/sinx) dx,then a2-a1,a3-a2,a4-a3........... are in

(A)AP
(B)GP
(C)HP
(D)none of these


Please show the working also!!!

Given a(n) = ∫(x = 0 to π/2) (sin^2(nx)/sin x) dx:

a(n+1) - a(n)
= ∫(x = 0 to π/2) (sin^2((n+1)x)/sin x) dx - ∫(x = 0 to π/2) (sin^2(nx)/sin x) dx
= ∫(x = 0 to π/2) [sin^2((n+1)x) - sin^2(nx)] dx / sin x
= ∫(x = 0 to π/2) [sin(nx+x) + sin(nx)] [sin(nx+x) - sin(nx)] dx / sin x
= ∫(x = 0 to π/2) [2 sin((2n+1)x/2) cos(x/2) * 2 sin(x/2) cos((2n+1)x/2)] dx / sin x
= ∫(x = 0 to π/2) [2 sin((2n+1)x/2) cos((2n+1)x/2) * 2 sin(x/2) cos(x/2)] dx / sin x
= ∫(x = 0 to π/2) sin((2n+1)x) * sin x dx / sin x
= ∫(x = 0 to π/2) sin((2n+1)x) dx
= -cos((2n+1)x)/(2n+1) {for x = 0 to π/2}
= 1/(2n+1).

This implies that {a(n+1) - a(n)} is in a harmonic progression.

I hope this helps!