If r is a positive quantity less than 1 and n is a positive integer , then prove that `(2n+1)r^(n) (1-r)lt 1 - r^(2n+1)`

If n be integer gt1, then prove that sum_(r=1)^(n-1) cos (2rpi)/n=-1

Prove that P(n,r) = (n- r+1) P(n,r-1)

Prove that ^(n-1) P_r+r .^(n-1) P_(r-1) = .^nP_r

Prove that .^(n-1) P_r+r .^(n-1) P_(r-1) = .^nP_r

If k a n d n are positive integers and s_k=1^k+2^k+3^k++n^k , then prove that sum_(r=1)^m^(m+1)C_r s_r=(n+1)^(m+1)-(n+1)dot

If R, n are positive integers, n is odd, 0lt F lt1" and if " (5sqrt(5)+11)^(n)=R+F, then prove that (R+F).F=4^(n)

If n is a positive integer then (1+x)^n=^nC_0 x^0+^nC_1 x^1+^nC_2^2+………+^nC_rx^r= sum _(r=0)^n ^nC_rx^r and (1-x)^n= ^nC_0x^0-^nC_1 x^1 +^nC_2-^nC_3x^3+………+(-1)^n ^nC_nx^n=sum_(r=0)^n (-1)^r ^nC_r x^r On the basis of above information answer the following question: If n is a positive integer then 1/((49)^n) - 8/((49)^n)(^(2n)C_1)+8^2/((49)^n)( ^(2n)C_2)- 8^3/((49)^n)(^(2n)c_3)+......+8^(2n)/((49)^n)= (A) -1 (B) 1 (C) (64/49)^n (D) none of these

Prove that: n(n-1)(n-2)....(n-r+1)=(n !)/((n-r)!)

If n is a positive integer then ^nC_r+^nC_(r+1)=^(n+1)^C_(r+1) Also coefficient of x^r in the expansion of (1+x)^n=^nC_r. In an identity in x, coefficient of similar powers of x on the two sides re equal. On the basis of above information answer the following question: If n is a positive integer then ^nC_n+^(n+1)C_n+^(n+2)C_n+.....+^(n+k)C_n= (A) ^(n+k+1)C_(n+2) (B) ^(n+k+1)C_(n+1) (C) ^(n+k+1)C_k (D) ^(n+k+1)C_(n-2)

If n is a positive integer and C_r = ""^nC_r then find the value of sum_(x = 1)^n r^2 ((C_r)/(C_(r - 1)) )