If `A_n` is the area bounded by `y=x` and `y=x^n ,n in \mathbb{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))`

Options

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))`

Answer

D

Explanation

`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))`