If `rArrI_(n)= int_(a)^(a+pi//2)(cos^(2)nx)/(sinx) dx, "then" I_(2)-I_(1),I_(3)-I_(2),I_(4)-I_(3)` are in

Options

A. G.P.
B. A.P.
C. H.P.
D. none of these

Answer

C

Explanation

We have,
`I_(n)-I_(n-1)= underset(a)overset(pi//2) int(cos^(2)nx-cos ^(2)(n-1)x)/(sinx)`
`rArrI_(n)-I_(n-1)= underset(a)overset(pi//2) int(sin^(2)(n-1)x-sin^(2)nx)/(sinx)dx`
`rArrI_(n)-I_(n-1)=- underset(a)overset(pi//2) intsin(2n-1)x dx`
`rArrI_(n)-I_(n-1)=(1)/(2n-1) [ cos (2n-1)x] _(0)^(pi//2)=-(1)/((2n-1))`
`:. I_(2)-I_(1)=-(1)/(3),I_(3)-I_(4),I_(4)=-(1)/(5),I_(4)-I_(3)=-(1)/(7)`
Clearly `,I_(2)-I_(3)-I_(4),I_(4)-I_(3)` are in H.P.