Question

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

Answer 1

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 × 2 × 3
i.e 8 < 6
Which is again not true.

for n = 4,
P(4) i.e 2⁴ < 4!
i.e 16 < 24
i.e a true statement.

P(5) : 2⁵ < 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