Let `A_n` be the area bounded by the curve `y=(tanx)^n` and the lines `x=0, y=0` and `x=pi/4`. Prove that for `n gt 2, A_n+A_(n-2)=1/(n-1)` and deduce that `1/(2n+2) lt A_n lt 1/(2n-2)`

Let A_n be the area bounded by the curve y=(tanx)^n and the lines x=0,y=0, and x=pi/4dot Prove that for n >2,A_n+A_(n-2)=1/(n-1) and deduce 1/(2n+2)

If A_n be the area bounded by the curve y=(tanx)^n and the lines x=0,\ y=0,\ x=pi//4 , then for n > 2.

Let A_n be the area bounded by the curve y = x^n(n>=1) and the line x=0, y = 0 and x =1/2 .If sum_(n=1)^n (2^n A_n)/n=1/3 then find the value of n.

If the area bounded by the curve y=cos^-1(cosx) and y=|x-pi| is pi^2/n , then n is equal to…

Prove that cos^(-1) ((1 - x^(2n))/(1 + x^(2n))) = 2 tan^(-1) x^(n), 0 lt x lt oo

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))

Let a_(0)=2,a_1=5 and for n ge 2, a_n=5a_(n-1)-6a_(n-2) . Then prove by induction that a_(n)=2^(n)+3^(n) forall n in Z^+ .

For each positive integer n gt a,A_(n) represents the area of the region restricted to the following two inequalities : (x ^(2))/(n ^(2)) + y ^(2) and x ^(2)+ (y ^(2))/(n ^(2)) lt 1. Find lim _(n to oo) A_(n).

If alphaa n dbeta are the rootsof he equations x^2-a x+b=0a n dA_n=alpha^n+beta^n , then which of the following is true? a. A_(n+1)=a A_n+b A_(n-1) b. A_(n+1)=b A_(n-1)+a A_n c. A_(n+1)=a A_n-b A_(n-1) d. A_(n+1)=b A_(n-1)-a A_n

A sequence of numbers A_n, n=1,2,3 is defined as follows : A_1=1/2 and for each ngeq2, A_n=((2n-3)/(2n))A_(n-1) , then prove that sum_(k=1)^n A_k<1,ngeq1