If `U_n=(sqrt(3)+1)^(2n)+(sqrt(3)-1)^(2n)` , then prove that `U_(n+1)=8U_n-4U_(n-1)dot`

If (4^(n+1).2^n-8^n)/(2^(3m))=3/8 then prove that n+1=m

Let u_(n)=(1)/(sqrt((5)))[((1+sqrt(5))/(2))^(n)-((1-sqrt(5))/(2))^(n)] (0=0,1,2,3,……) , prove that u_(n+1)=u_(n)+u_(n-1)(n ge 1) .

If u_(n)=int_(0)^((pi)/(2))theta sin^(n)theta d theta and n>=1, then prove that u_(n)=((n-1)/(n))u_(n-2)+(1)/(n^(2))

If U_n=int_0^(pi/2)(sin^2n x)/(sin^2x)dx, then show that U_1,U_2,U_3.......U_n constitute an AP. Hence or otherwise find the value of U_n.

lim_(n rarr4)(sqrt(2n+1)-3)/(sqrt(n-1)-sqrt(2))

If U_(n)=2cos n theta, then U_(1)U_(n)-U_(n-1) is equal to -

If u_(n)=sin^(n)theta+cos^(n)theta, then prove that (u_(5)-u_(7))/(u_(3)-u_(5))=(u_(3))/(u_(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) .