If `P(n)=2+4+6+....+2n, n in N`. Then `P(k) =k(k+1)` `rArr P(k+1)=(k+1)(k+2),forall k in N` , So , we can conclude that `P(n)=n(n+1)` for

If P(n)=2+4+6+…..+2n ninN then P(k) =k(k+1) implies P(k+1)=(k+1)(k+2) for all kinN . So we can conclude that P(n)=n(n+1) for:

If P (n) = 2 + 4 + 6+…+ 2n, n in N then P(k) = k (k +1) + 2 implies P (k +1 ) = (k +1 ) (k + 2)+ 2 AA k in N. So, we can conciude that P (n) = n (n +1) + 2 for

If P(n) is a statement (n in N) such that if P(k) is true,P(k+1) is true for k in N, then P(k) is true.

Let P(n) be a statement such that P(n) Rightarrow P(n+1) for all n in NN . Also P(k) is true, k in NN . Then we can conclude that:

Let P(n) be a statement such that P(n) Rightarrow P(n+1) for all n in NN . Also, if P(k) is true, k in N , then we can conclude that.-

Let P(n) be statement and let P(k) rArr P(k+1) ,for some natural number k, then P(n) is true for all n in N

If…..is true and P(k) is true rArr P(k+1) is true, k gt= - 1 , then for all n in N cup {0,-1},P(n) is true.

lim_ (n rarr oo) (1 ^ (k) + 2 ^ (k) + ... + n ^ (k)) / (k * n ^ (k + 1)), (n in N, k in I ^ (+))

If a_k=1/(k(k+1)) for k=1 ,2……..,n then prove that (sum_(k=1)^n a_k)^2 =n^2/(n+1)^2