Show that `(n^5)/(5) + (n^3)/(3) + (7n)/(15)` is a natural number for all `n in N`

Show that 12^(n) cannot end with digit 0 or 5 for any natural number n.

Prove that , (1)/(n+1) + (1)/(n+2) + ………….+ (1)/(2n) gt 13/24 for all natural numbers n > 1

Let n(A) = n, then the number of all relations on A, is

Use the principle of mathematical induction to show that 5^(2n+1)+3^(n+2).2^(n-1) divisible by 19 for all natural numbers n.

Show that 3^(n) xx 4^(m) cannot end with the digit 0 or 5 for any natural numbers ‘n’and 'm'

Use the principle of mathematical induction : A sequence b_0,b_1,b_2 ,….. Is defined by letting b_0 = 5 and b_k = 4+b_(k-1) , for all natural numbers k. show that b_n = 5 + 4n , for all natural number n using mathematical induction.

P(n) : n (n+1) is even number then P(3) = .........

Show that the sequence t_(n) defined by t_(n)=(2^(2n-1))/(3) for all values of n in N is a GP. Also, find its common ratio.

Prove each of the statements by the principle of mathematical induction : 1+5+9+……+(4n-3)= n(2n-1) for all natural numbers n .