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 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
$\therefore F(n): 2^{n}
$F(n): 2^{n}