Question

Prove by induction that:


For all natural numbers n, (n^3 / 3)+(n^5 /5)+(7n/ 15) is an integer

Answer 1

This can be proved by the principle of mathematical induction. For n=1n=1, we can see that the result is a number ∈N∈N. Assume it to be true for k∈Nk∈N. Then,

k5/5+k3/3+7k/15=pk5/5+k3/3+7k/15=p, where p∈Np∈N.

Now, we have to prove it for k+1k+1.

(k+1)55+(k+1)33+7(k+1)15(k+1)55+(k+1)33+7(k+1)15 can be expanded using the binomial theorem. Segregate the terms so obtain pp as a term in the equation. The other terms with like denominators can be added, and the denominators will be cancelled out, leaving a result like p+qp+q, where q∈Nq∈N. Therefore, p+q∈Np+q∈N. Hence, proved