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Question: If a<sub>n</sub> = \(\int _ { 0 } ^ { \pi / 2 } \frac { \sin ^ { 2 } \mathrm { nx } } { \sin \mathr...

If an = ∫0π/2sin⁡2nxsin⁡x\int _ { 0 } ^ { \pi / 2 } \frac { \sin ^ { 2 } \mathrm { nx } } { \sin \mathrm { x } } dx then a2 – a1, a3 – a2, a4–a3,......are in –

A

AP

B

GP

C

HP

D

None of these

Answer

HP

Explanation

Solution

an – an–1 = ∫0π/2sin⁡2(nx)sin⁡x\int _ { 0 } ^ { \pi / 2 } \frac { \sin ^ { 2 } ( n x ) } { \sin x } –

= 12\frac { 1 } { 2 }

= ∫0π/2sin⁡(2n−1)xdx=\int _ { 0 } ^ { \pi / 2 } \sin ( 2 n - 1 ) x d x = [−cos⁡(2n−1)x(2n−1)]0π/2\left[ - \frac { \cos ( 2 n - 1 ) x } { ( 2 n - 1 ) } \right] _ { 0 } ^ { \pi / 2 }=

Ž a2 – a1 = 13\frac { 1 } { 3 }, a3 – a2 = 15\frac { 1 } { 5 }, a4 – a3=17\frac { 1 } { 7 }..... are in HP