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

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

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