If P(n) : 2n < n!, n ∈ N,
Question:

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

Solution:

Given P(n) : 2n < n! ; n ∈ N

for n = 1,

P(1) : 2′ < 1!

i.e 2 < 1

Which is not true

for n = 2,

P(2) : 22 = 4 < 2!

i.e 4 < 2

Which is not true

for n = 3,

P(3) : 23 < 3!

i.e $8<1 \times 2 \times 3$

 

i.e $8<6$

Which is again not true

for n = 4,

P(4) i.e 2< 4!

i.e 16 < 24

i.e a true statement.

P(5) : 2< 5!

i.e 32 < 1 × 2 × 3 × 4 × 5

i.e 32 < 120

Which is also true

∴ P(n) : 2n < n! is true for n > 3

P(n) : 2n < n! is true i.e for n ≥ 4

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