If `a_1,a_2,……….,a_(n+1)` are in A.P. prove that `sum_(k=0)^n ^nC_k.a_(k+1)=2^(n-1)(a_1+a_(n+1))`

If a_(1),a_(2),a_(3),…a_(n+1) are in arithmetic progression, then sum_(k=0)^(n) .^(n)C_(k.a_(k+1) is equal to

"If "a_1,a_2,a_3,.....,a_n" are in AP, prove that "a_(1)+a_(n)=a_(r)+a_(n-r+1)""

If a_(1),a_(2),a_(3)...a_(2n-1) are in H.P.then sum_(k=1)^(2n)(-1)^(k)(a_(k)+a_(k+1))/(a_(k)-a_(k+1)) is equal to

If a_1,a_2,a_3 are in G.P. having common ratio r such that sum_(k=1)^n a_(2k-1)= sum_(k=1)^na_(2k+2)!=0 then number of possible value of r is (A) 1 (B) 2 (C) 3 (D) none of these

If a_(1),a_(2),a_(3),......,a_(n) are in A.P. Where a_(k)>0AA K and sum_(K=2)^(n)(1)/(sqrt(a_(K-1))+sqrt(a_(K)))=(L)/(sqrt(a_(1))+sqrt(a_(n))) then L=

If a_1,a_2,a_3,.....,a_n are in AP, prove that 1/(a_1a_2)+1/(a_2a_3)+1/(a_3a_4)+...+1/(a_(n-1)a_n)=(n-1)/(a_1a_n) .

If a_1,a_2,a_3….a_(2n+1) are in A.P then (a_(2n+1)-a_1)/(a_(2n+1)+a_1)+(a_2n-a_2)/(a_(2n)+a_2)+....+(a_(n+2)-a_n)/(a_(n+2)+a_n) is equal to

If a_1,a_2, a_3………. are in H.P. and f(k) =sum_(r=1)^n a_r-a_k, the a_1/(f(1)), a_2/(f(2)), a_3/(f(3)), …….., a_n/(f(n)) are in (A) A.P. (B) G.P (C) H.P. (D) none of these

Prove by binomial expansion that sum_(k=1)^nk^2*"^nC_k=n(n+1)2^(n-2)