prove that `1P_1+2.2P_2+3.3P_3+.........+n.nP_n=(n+1)P_(n+1)-1`

A coin has probability p of showing head when tossed. It is tossed n times. Let P_n denote the probability that no two (or more) consecutive heads occur. Prove that P_1 = 1,P_2 = 1 - p^2 and P_n= (1 - p) P_(n-1) + p(1 - p) P_(n-2) for all n geq 3 .

If P_m stands for ^m P_m , then prove that: 1+1. P_1+2. P_2+3. P_3++ndotP_n=(n+1)!

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

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

Prove that: P(1,1)+2. P(2,2)+3. P(3,3)++ndotP(n , n)=P(n+1,\ n+1)-1.

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

Prove that 1+1* ""^(1)P_(1)+2* ""^(2)P_(2)+3* ""^(3)P_(3) + … +n* ""^(n)P_(n)=""^(n+1)P_(n+1).

If P_n is the sum of a GdotPdot upto n terms (ngeq3), then prove that (1-r)(d P_n)/(d r)=(1-n)P_n+n P_(n-1), where r is the common ratio of GdotPdot

If P_n is the sum of a GdotPdot upto n terms (ngeq3), then prove that (1-r)(d P_n)/(d r)=(1-n)P_n+n P_(n-1), where r is the common ratio of GdotPdot

Prove that: (i) (.^(n)P_(r))/(.^(n)P_(r-2)) = (n-r+1) (n-r+2)