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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. |
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Answer» 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. |
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