is a natural number for all n∈N. Step I : For n=1,P(1):‌
1
5
+‌
1
3
+‌
7
15
=1∈N Hence, it is true for n=1. Step II : Let it is true for n=k, i.e. ‌
k5
5
+‌
k3
3
+‌
7k
15
=λ∈N . . . (i) ‌‌ Step III : For ‌n=k+1 ‌‌
(k+1)5
5
+‌
(k+1)3
3
+‌
7(k+1)
15
‌=‌
1
5
(k5+5k4+10k3+10k2+5k+1) +‌
1
3
(k3+3k2+3k+1)+‌
7
15
k+‌
7
15
=(‌
k5
5
+‌
k3
3
+‌
7
15
k)+(k4+2k3+3k2+2k) +‌
1
5
+‌
1
3
+‌
7
15
=λ+k4+2k3+3k2+2k+1 [using equation (i)] which is a natural number, since λk∈N. Therefore, P(k+1) is true, when P(k) is true, Hence, from the principle of mathematical induction, the statement is true for all natural numbers n.