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

Explanation

For n=1
L.H.S `=2xx1+7=9`
R.H.S. `=(1+3)^(2)=16`
`:. " "L.H.S. lt R.H.S.`
`rArr` Given statement is true for n=1
Let given statement be true for n=k
`:. " "2k+7 lt (k+3)^(2)`
for n=K+1
`2(k+1)+7 =(2k+7)+2`
`lt (k+3)+2`
[From inequation (1)]
`=K^(2)+6k+11`
`lt (k^(2)+6K+11)+(2k+5)`
`lt K^(2)+8K+16lt (k+4)^(2)`
` rArr 2(k+1) +7 lt (K+4)^(2)`
`rArr` Given statement is also true for n=K+1
Hence from the principle of mathematical induction P (n) is true for all natural numbes n.