Prove that :
` P(n,n)= 2P (n,n -2) `

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

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

Prove the following: P(n , n)=2P(n , n-2)

Prove the following: 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.

If P(n) is the statement n^2-n+41 is prime. Prove that P(1),\ P(2)a n d\ P(3) are true. Prove also that P(41) is not true.

Prove the following: P(n , r)=P(n-1, r)+rdotP(n-1,\ r-1)

If P be the sum of odd terms and Q be the sum of even terms in the expansion of (x+a)^n , prove that (iii) 2(P^2 + Q^2 ) = (x+a)^(2n) + (x-a)^(2n) .

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

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