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Prove the following by using the princip...

Prove the following by using the principle of mathematical induction for all `n in N`:`(2n+7)<(n+3)^2`.

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`P(n) : (2n+7)lt (n+3)^(2)` for all `n in N`
For `n = 1`
L.H.S. `= 2 xx 1 + 7 = 9`
and `R.H.S. = (1+3)^(2) = 16`
L.H.S. `lt R.H.S.`
So, `P(1)` is true.
Let `P(k)` be true.
i.e, `(2k+7) lt (k+3)^(2)"….."(1)`
We have to prove that ` P(k+1)` is true.
i.e, `(2(k+1) + 7) lt (k+1+3)^(2)`
or `(2k+9) lt (k+4)^(2)`
Now, `(2(k+1)+7)`
`= (2k+7) + 2 lt (k+3)^(2) + 2 "......"(2)` [Using `(1)`]
Now, `(k+3)^(2) + 2 - (k+4)^(2)`
`= k^(2) + 6k + 9 + 2 - k^(2) - 8k - 16`
`= - 2k - lt 0`
`:. (2(k+1)+7) lt (k+4)^(2)`
Thus, `P(k+1)` is true whenever `P(k)` is true.
Hence, by the principle of mathematical induction, statement `P(n)` is true for all natural numbers.
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