Q29 of 30 Page 70

Fill in the blanks in the following:

If P(n) : 2n < n!, n ϵ N, then P(n) is true for all n ≥ __________.

Given; P(n): 2n < n! n ϵ N

⇒ P(1) : 2×1<1! ⇒ 2<1; it’s not true.


⇒ P(2) : 2×2<2! ⇒ 4<2; it’s not true.


⇒ P(3) : 2×3<3! ⇒ 6<6; it’s not true.


⇒ P(4) : 2×4<4! ⇒ 8<24; it’s true.


⇒ P(3) : 2×5<5! ⇒ 10<120; it’s true.


∴ n≥4


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True or False

Let P(n) be a statement and let P(k) ⟹ P(k + 1), for some natural number k, then P(n) is true of all n ϵ N.