If `A_n` be the area bounded by the curve `y=(tanx^n)` ands the lines `x=0,\ y=0,\ x=pi//4` Prove that for `n > 2.` , `A_n+A_(n+2)=1/(n+1)` and deduce `1/(2n+2)< A_(n) <1/(2n-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…

If A_(n) is the area bounded by y=(1-x^(2))^(n) and coordinate axes,n in N, then

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

If A_(n) is the area bounded by y=x and y=x^(n),n in N, then A_(2)*A_(3)...A_(n)=(1)/(n(n+1)) (b) (1)/(2^(n)n(n+1))(1)/(2^(n-1)n(n+1)) (d) (1)/(2^(n-2)n(n+1))

If I_(n)=int_(0)^(1)x^(n)(tan^(-1)x)dx, then prove that(n+1)I_(n)+(n-1)I_(n-2)=-(1)/(n)+(pi)/(2)

If a_(0), a_(1), a(2),… are the coefficients in the expansion of (1 + x + x^(2))^(n) in ascending powers of x, prove that a_(0) a_(2) - a_(1) a_(3) + a_(2) a_(4) - …+ a_(2n-2) a_(2n)= a_(n+1) .

If y^(2)=ax^(2)+2bx+c , and u_(n)= int (x^(n))/(y)dx , prove that (n+1)a u_(n+1)+(2n+1)bu_(n)+(n)c u_(n-1)=x^(n)y and deduce that au_(1)=y-b u_(0), 2a^(2)u_(2)=y(ax-3b)-(ac-3b^(2))u_(0) .

Let {A_n} be a unique sequence of positive integers satisfying the following properties: A_1 = 1, A_2 = 2, A_4 = 12, and A_(n+1) . A_(n-1) = A_n^2 pm 1 for n = 2,3,4… then , A_7 is