If `T_r` is the rth term in the expansion of `(1+x)^n` in ascending powers of x. then show that `r(r+1)T_(r+2) = (n-r+1)(n-r)x^2T_r`

If t_r is the rth term is the expansion of (1 + a)^n , in ascending powers of a, prove that : r(r+1) t_(r+2)= (n-r+1)(n-r) a^2 t_r .

If k_r is the coefficient of y^(r-1) in the expansion of (1+2y)^10 , in ascending powers of y, determine r when (k_(r +2))/k_r=4 .

If in the expansion of (1-x)^(2n-1) , the coefficient of x^r is denoted by a_r , then show that a_(r-1)+a_(2n-r)=0 .

If the coefficient of the rth, (r+1)th and (r+2)th terms in the expansion of (1+x)^(n) are in A.P., prove that n^(2) - n(4r +1) + 4r^(2) - 2=0 .

If the coefficients of rth, (r+1)t h ,a n d(r+2)t h terms in the expansion of (1+x)^(14) are in A.P., then r is/are a. 5 b. 11 c. 10 d. 9

Find the coefficient of x^(n-r) in the expansion of (x+1)^n (1+x)^n . Deduce that C_0C_r+C_1C_(r-1)+......+C_(n-r) C_n= ((2n!))/((n+r)!(n-r)!) .

In the binomial expansion of (1+x)^34 , the coefficients of the (2r-1)th and the (r-5)th terms are equal. Find r.

The coefficients of the (r-1)^(th) , r^(th) and (r+1)^(th) terms in the expansion of (x + 1)^n are in the ratio 1:3:5. Find n and r.