prove that `n^(2)lt2^(n)`, for all natural number `n≥5`.

Explanation

Consider the given statement
`P(n):n^(2)ltn^(n)` for all natural number `nle5`.
Step I We observe that P(5) is true `P(5):5^(2)lt2^(5)`
`=25lt32`
Hence, P(5) is ture.
Step II Now, assume that P(n) true for n=k.
`P(k)=k^(2)lt2^(k)` is true.
Step III Now, to prove P(k+1) is true, we have to show that
`P(k+1):(k+1)^(2)lt2^(k+1)`
Now, `k^(2)lt2^(k)=k^(2)+2k+1lt2^(k)+2k+1` . . . (i)
`=(k+1)^(2)lt2^(k)+2k+1`
Now, `(2k+1)lt2^(k) =2^(k)+2k+2lt2^(k)+2^(k)`
`=2^(k)+2k+1lt2*2^(k)`
`=2^(k)+2k+1lt2^(k=1)` . . . (ii)
Form Eqs. (i) and (ii), we get `(k+1)^(2)lt2^(k+1)`
So, P(k+1) is true, whenever P(k) is true. Hence, by the principle of mathematical induction P(n) is true for all natural numbers `nle5`.