Prove that `^10P_3=^9P_3+3.^9P_2`

Prove that P(10,3)=P( 9,3) +3P ( 9,2)

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

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

If the roots of the equation x^(3) - px^(2) + qx - r = 0 are in A.P., then prove that, 2p^3 −9pq+27r=0

If p_(1),p_(2),p_(3) are the perpendiculars from the vertices of a triangle to the opposite sides, then prove that p_(1)p_(2)p_(3)=(a^(2)b^(2)c^(2))/(8R^(3))

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 .

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

If p=2-a , prove that a^3+6a p+p^3-8=0

If p=2-a , prove that a^3+6a p+p^3-8=0