Question:
If P(n) : n2 < 2n, n ∈ N, then P(n) is true for all n ≥ _____________.
Solution:
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 $n=7, P(7): 49<2^{7}=128$ which is again true statement
Hence, P(n) : n2 < 2n is true for n ≥ 5
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