If `l_(n)=int_(0)^((pi)/(4)) tan^(n) xdx` show that `(1)/(l_(2)+l_(4)),(1)/(l_(3)+l_(5)),(1)/(l_(4)+l_(6)),(1)/(l_(5)+l_(7)),"...."` from an AP. Find its common difference.

Explanation

We have, `l_(n)+l_(n+2)=int_(0)^((pi)/(4)) (tan^(n)x+tan^(n+2)x) dx`
`=int_(0)^((pi)/(4))tan^(n)x(1+tan^(2)x) dx`
`=int_(0)^((pi)/(4))tan^(n)x*sec^(2)xdx=[(tan^(n+1)x)/(n+1)]_(0)^((pi)/(4))=(1)/(n+1)`
Hence, `(1)/(l_(n)+l_(n+2))=n+1`
On putting `n=2,3,4,5"...."`
`:.(1)/(l_(2)+l_(4))=3,(1)/(l_(3)+l_(5))=4,(1)/(l_(4)+l_(6))=5,(1)/(l_(5)+l_(7))=6,"...."`
Hence, `(1)/(l_(2)+l_(4)),(1)/(l_(3)+l_(5)),(1)/(l_(4)+l_(6)),(1)/(l_(5)+l_(7)),"...."` are in AP with common difference1.