Show that the middle term in the expansion of `(1 + x)^(2n)` is `(1.3.5.........(2n - 1))/(n!)2^(n)x^(n)`, where n is a positive integer.

Prove that 2^n >1+nsqrt(2^(n-1)),AAn >2 where n is a positive integer.

If the middle term in the expansion of (x^2+1//x)^n is 924 x^6, then find the value of ndot

In the expansion of (1 + x)^(n) (1 + y)^(n) (1 + z)^(n) , the sum of the co-efficients of the terms of degree 'r' is

Show that the coefficient of the middle term in the expansion of (1+x)^2n is equal to the sum of the coefficients of two middle terms in the expansion of (1 + x)^2n-1

If A=[(3,-4),(1,-1)] , then prove that A^(n)=[(1+2n,-4n),(n,1-2n)] , where n is any positive integer.

Prove that the coefficient of x^n in the expansion of 1/((1-x)(1-2x)(1-3x))i s1/2(3^(n+2)-2^(n+3)+1)dot

If the 4th term in the expansion of (a x+1//x)^n is 5/2, then (a) a=1/2 b. n=8 c. a=2/3 d. n=6

Let m be the smallest positive integer such that the coefficient of x^2 in the expansion of (1+x)^2 + (1 +x)^3 + (1 + x)^4 +........+ (1+x)^49 + (1 + mx)^50 is (3n + 1) .^51C_3 for some positive integer n. Then the value of n is

If C_r stands for nC_r , then the sum of the series (2(n/2)!(n/2)!)/(n !)[C_0^2-2C_1^2+3C_2^2-........+(-1)^n(n+1)C_n^2] ,where n is an even positive integer, is