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 1P_1+2.2P_2+3.3P_3+.........+n.nP_n=(n+1)P_(n+1)-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 P(n,n) = P(n,n-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 in an A.P, S_n=n^2p and S_m=m^2p , then prove that S_p is equal to p^3

If P(n) be the statement " 10n+3 is a prime number", then prove that P(1) and P(2) are true but P(3) is false.

Given four points P_1,P_2,P_3a n dP_4 on the coordinate plane with origin O which satisfy the condition ( vec (O P))_(n-1)+( vec(O P))_(n+1)=3/2 vec (O P)_(n) . If P1 and P2 lie on the curve xy=1 , then prove that P3 does not lie on the curve

If (1-x+x^(2))^(4)=1+P_(1)x+P_(2)x^(2)+P_(3)x^(3)+...+P_(8)x^(8) , then prove that : P_(2)+P_(4)+P_(6)+P_(8)=40 and P_(1)+P_(3)+P_(5)+P_(7)=-40 .

A is a set containing n elements. A subset P_1 is chosen and A is reconstructed by replacing the elements of P_1 . The same process is repeated for subsets P_1,P_2,....,P_m with m>1 . The number of ways of choosing P_1,P_2,....,P_m so that P_1 cup P_2 cup....cup P_m=A is (a) (2^m-1)^(mn) (b) (2^n-1)^m (c) (m+n)C_m (d) none of these

If P_1, P_2, …. P_n are points lying on the bisector of first quadrant and OP_1 = 1, OP_n = n. OP_(n-1) , n=2, 3,4,… and P_n is (2520sqrt(2), 2520sqrt(2)) , then n= .