If `sum_(r=0)^(2n)a_r(x-2)^r=sum_(r=0)^(2n)b_r(x-3)^ra n da_k=1` for all `kgeqn ,` then show that `b_n=^(2n+1)C_(n+1)` .

If sum_(r=0)^n{a_r(x-alpha+2)^r-b_r(alpha-x-1)^r}=0, then prove that b_n-(-1)^n a_n=0.

If sum_(r=0)^nr/(^n C_r)=sum_(r=0)^n(n^2-3n+3)/(2.^n C_r) , then a. n=1 b. n=2 c. n=3 d. none of these

Statement1: if n in Na n dn is not a multiple of 3 and (1+x+x^2)^n=sum_(r=0)^(2n)a_r x^r , then the value of sum_(r=0)^n(-1)^r a r^n C_r is zero Statement 2: The coefficient of x^n in the expansion of (1-x^3)^n is zero, if n=3k+1orn=3k+2.

If (4x^(2) + 1)^(n) = sum_(r=0)^(n)a_(r)(1+x^(2))^(n-r)x^(2r) , then the value of

If sum_(r=0)^(n) (pr+2).""^(n)C_(r)=(25)(64) where n, p in N , then

If s_(n) = sum_(r = 0)^(n) 1/(""^(n)C_(r)) and t_(n) = sum _(r = 0)^(n) r/(""^(n)C_(r)) ,then t_(n)/s_(n) is equal to

The value of sum_(r=0)^(3n-1)(-1)^r 6nC_(2r+1)3^r is

If (1+2x+x^2)^n=sum_(r=0)^(2n)a_r x^r ,then a= (^n C_2)^2 b. ^n C_rdot^n C_(r+1) c. ^2n C_r d. ^2n C_(r+1)

If sum_(r=0)^(n){("^(n)C_(r-1))/('^(n)C_(r )+^(n)C_(r-1))}^(3)=(25)/(24) , then n is equal to

Find the sum of sum_(r=1)^n(r^n C_r)/(n C_(r-1) .