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If A(n) is the area bounded by y=x and y...

If `A_(n)` is the area bounded by y=x and `y=x^(n), n in N,` then `A_(2).A_(3)…A_(n)=`

A

`(1)/(n(n+1))`

B

`(1)/(2^(n)n(n+1))`

C

`(1)/(2^(n-1)n(n+1))`

D

`(1)/(2^(n-2)n(n+1))`

Text Solution

Verified by Experts

The correct Answer is:
D
`y=x" intersect "y=x^(n)" at "(0,0) and (1,0)" for all " n in N`

`"Area "A_(n)=overset(1)underset(0)int(x-x^(n))dx`
`=[(x^(2))/(2)-(x^(n+1))/(n+1)]_(0)^(1)`
`=(1)/(2)-(1)/(n+1)`
`=(n-1)/(2(n+1))`
`"Thus, "A_(2)cdotA_(3)cdotA_(4)....A_(n)=(1)/(2^(n-1))((1)/(3)cdot(2)/(4)cdot(3)/(5)...(n-1)/(n+1))`
`=(1)/(2^(n-2).n(n+1))`
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