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If I(n) = int(0)^(n) ( 1- sin 2nx)/( 1-c...

If `I_(n) = int_(0)^(n) ( 1- sin 2nx)/( 1-cos 2n)dx` then `I_(1) , I_(2) , I_(3),"........"` are in `:`

A

AP

B

GP

C

HP

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
`:. I_(n)=int_(0)^(pi)(1-sin2nx)/(1-cos2x)dx " " implies I_(n)=int_(0)^(pi)(1-sin2nx)/(2sin^(2)x)dx`
`implies I_(n+1)+I_(n-1)-2 I_(n) [ 1-sin2 (n+1)x+1-sin 2(n-1)x-2]`
`=(1)/(2)int_(0)^(pi)(+sin2nx " "]/(sin^(2)x)dx`
`=(1)/(2)int_(0)^(pi)([sin2nx-sin 2 (n+1)x]+[sin2nx-sin 2 (n-1)x])/(sin^(2)x)dx`
`=(1)/(2)int_(0)^(pi)(-2cos(2n+1)xsin(x)+2cos(2n-1)xsinx)/(sin^(2)x)dx`
`=int_(0)^(pi)(sinx[cos(2n-1)x-cos(2n+1)x])/(sin^(2)x)dx`
`=int_(0)^(pi)(2sin2nxsinx)/(sinx)dx=2int_(0)^(pi)sin2nxdx=(2)/(2n)[-cos2nx]_(0)^(pi)`
`=-(1)/(n)(1-1)=0` `:. I_(n+1)+I_(n-1)=2I_(n)`
`implies I_(n-1)+I_(n),I_(n+1)` are in AP.
`:.I_(1),I_(2),I_(3)"...."` are in AP.
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