Let P(n) = (2n + 7) < (n+3)2 P(1) is true Let P(k) = (2k+7) < (k+3)2 To Prove P(k+1) = (2(k+1)+7) < (k + 4)2 = [2k+9] < (k+4)2 = [(k+3)2 – 9] < (k + 4)2 = [( k+3+3) (k+3-3)] < ( k+4)2 = [(k+6)(k) < (k+4)2 = (k2 + 6k) < (k2 + 8k + 16) The given expression is true for P(k+1)

Q24 of 24 Page 4

(2n + 7) < (n+3)².

Let P(n) = (2n + 7) < (n+3)²
P(1) is true
Let P(k) = (2k+7) < (k+3)²
To Prove P(k+1) = (2(k+1)+7) < (k + 4)²= [2k+9] < (k+4)²= [(k+3)² – 9] < (k + 4)²= [( k+3+3) (k+3-3)] < ( k+4)²= [(k+6)(k) < (k+4)²= (k² + 6k) < (k² + 8k + 16)
The given expression is true for P(k+1)