Question

If P(n) : n² < 2ⁿ, n ∈ N, then P(n) is true for all n ≥ _____________.

Answer 1

P(n) : n² < 2ⁿ, n ∈ N

for n = 1,
P(1) : 1 < 2 which is true statement

for n = 2,
P(2) : 2² < 2² which is not true/false statement

for n = 3,
P(3) : (3)² < 2³
i.e 9 < 8 which is a false statement

for n = 4,
P(4) : (4)² < 2⁴
i.e 16 < 4 × 4 = 16
which is a false statement

for n = 5,
P(5) : (5)² < 2⁵
i.e 25 < 32
which is a true statement

for n = 6,
P(6) : (36) < 2⁶
i.e 36 < 4 × 4 × 4 = 64
Which is a true statement

for n = 7, P(7) : 49 < 2⁷ = 128 which is again true statement
Hence, P(n) : n² < 2ⁿ is true for n ≥ 5