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

Prove the following: P(n , n)=P(n , n-1)

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

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

Prove the following by using the principle of mathematical induction for all n in N : a+a r+a r^2+...+a r^(n-1)=(a(r^n-1))/(r-1)

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

If ((n),(r )) "denotes " ""^nC_r then (a) Evalutae : 2^(15)((30),(0))((30),(1))-2^(14)((30),(1))((29),(14))+2^(13)((30),(2))((28),(13)).......-((30),(15))((15),(0)) ( b) Prove that : Sigma_(r=1)^(n) ((n-1),(n-r))((n),(r))=((2n-1),(n-1)) ( c) Prove that : ((n),(r))((r),(k))=((n),(k))((n-r),(r-k))

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

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) r.^(n)C_(r) =(n-r+1).^(n)C_(r-1) (ii) n.^(n-1)C_(r-1) = (n-r+1) .^(n)C_(r-1) (iii) .^(n)C_(r)+ 2.^(n)C_(r-1) +^(n)C_(r-2) =^(n+2)C_(r) (iv) .^(4n)C_(2n): .^(2n)C_(n) = (1.3.5...(4n-1))/({1.3.5..(2n-1)}^(2))