Prove by the principle of mathematical induction that `(n^5)/5+(n^3)/3+(7n)/(15)` is a natural number for all `n in Ndot`

Explanation

P(n):`n^5/5+n^3/3+(7n)/15` is a natural number.

`n^5/5+n^3/3+(7n)/15=lambda`

`P(1):1/5+1/3+7/15=(3+5+7)/15=1`

`P(m):m^5/5+m^3/3+(7m)/15=lambda`

`P(m+1):(m+1)^5/5+(m+1)^3/3+(m(m+1))/15=k`

`=1/5(m^5+5m^4+10m^3+10m^2+5m+1)+1/3(m^3+3m^2+3m+1)+(7m)/15+1/15`

`=(m^5/5+m^3/3+(7m)/k)+(m^4+m^3+3m^2+2m)+1/5+1/3+7/15`

`=lambda+m^4+2m^3+3m^2+2m+1`

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