Question -
Answer -
We can write the givenstatement as
P(n): (2n +7)< (n + 3)2
If n = 1 we get
2.1 + 7 = 9 < (1 +3)2 = 16
Which is true.
Consider P (k) be truefor some positive integer k
(2k + 7)< (k + 3)2 … (1)
Now let us prove thatP (k + 1) is true.
Here
{2 (k + 1) + 7} = (2k+ 7) + 2
We can write it as
= {2 (k + 1) + 7}
From equation (1) weget
(2k + 7) + 2 < (k +3)2 + 2
By expanding the terms
2 (k + 1) + 7 < k2 +6k + 9 + 2
On further calculation
2 (k + 1) + 7 < k2 +6k + 11
Here k2 +6k + 11 < k2 + 8k + 16
We can write it as
2 (k + 1) + 7 < (k+ 4)2
2 (k + 1) + 7 < {(k+ 1) + 3}2
P (k + 1) is truewhenever P (k) is true.
Therefore, by theprinciple of mathematical induction, statement P (n) is true for all naturalnumbers i.e. n.