(2n + 7) < (n + 3)2

1.	(2n + 7) < (n + 3)2
Answer» Let P(n): (2n + 7) < (n + 3)² For = 1, (2 + 7) < (1+ 3)² ⇒9<16 is true. ∴ P(1) is true. Let us assume P(k) is true for some k∈ N i.e., (2k+ 1) < (k + 3)² Consider 2 (k + 1) + 7 = 2k + 2 + 7 = (2k + 7) + 2 < (k: + 3)² + 2 = k² + 6k + 9 + 2 <(k + 4)² ∴ P(k +1) is true. Hence, by mathematical induction, P(n) is true for all n∈N

Answer»

Let P(n): (2n + 7) < (n + 3)²

For = 1, (2 + 7) < (1+ 3)² ⇒9<16 is true.

∴ P(1) is true.

Let us assume P(k) is true for some k∈ N

i.e., (2k+ 1) < (k + 3)²

Consider 2 (k + 1) + 7 = 2k + 2 + 7

= (2k + 7) + 2 < (k: + 3)² + 2 = k² + 6k + 9 + 2 <(k + 4)²

∴ P(k +1) is true.

Hence, by mathematical induction, P(n) is true for all n∈N

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