Divide :`(a^(2^n)-b^(2 ^n))`by `(a^(2^(n-1))+b^(2^(n-1)))`

(x^(2^(n-1))+y^(2^(n-1)))(x^(2^(n-1))-y^(2^(n-1)))=

lim_(n->oo)((n^2-n+1)/(n^2-n-1))^(n(n-1)) is

Find the sum of the series upto n terms ((2n+1)/(2n-1))+3((2n+1)/(2n-1))^2+5((2n+1)/(2n-1))^3+.....

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))

Two players P_1a n dP_2 play a series of 2n games. Each game can result in either a win or a loss for P_1dot the total number of ways in which P_1 can win the series of these games is equal to a. 1/2(2^(2n)-.^(2n)C_n) b. 1/2(2^(2n)-2xx.^(2n)C_n) c. 1/2(2^n-.^(2n)C_n) d. 1/2(2^n-2xx.^(2n)C_n)

The coefficient of 1//x in the expansion of (1+x)^n(1+1//x)^n is (a). (n !)/((n-1)!(n+1)!) (b). ((2n)!)/((n-1)!(n+1)!) (c). ((2n)!)/((2n-1)!(2n+1)!) (d). none of these

If the expansion in powers of x of the function 1/[(1-a x)(1-b x)] is a a_0+a_1x+a_2x^2+a_3x^3+ ,t h e na_n i s a. (b^n-a^n)/(b-a) b. (a^n-b^n)/(b-a) c. (b^(n+1)-a^(n+1))/(b-a) d. (a^(n+1)-b^(n+1))/(b-a)

Prove that by using the principle of mathematical induction for all n in N : 1^(2)+3^(2)+5^(2)+...(2n-1)^(2)= (n(2n-1)(2n+1))/(3)

Evaluate lim_(ntooo) n^(-n^(2))[(n+2^(0))(n+2^(-1))(n+2^(-2))...(n+2^(-n+1))]^(n) .

lim_(nrarroo) [(1)/(n)+(sqrt(n^(2)-1^(2)))/(n^(2))+(sqrt(n^(2)-2^(2)))/(n^(2))+...+(sqrt(n^(2)-(n-1)^(2)))/(n^(2))]