n^2 < 2^n for all natural numbers: - Vidyadip

MCQ

$n^2 < 2^n$ for all natural numbers:

✓
$n ≥ 5$
B
$n < 5$
C
$n > 1$
D
$n ≤ 3$

✓

Answer

Correct option: A.

$n ≥ 5$

Consider, $P(n) : n^2 < 2^n$
Substituting $n = 1, 2, 3,…$
$P(1): 1^2< 2^1$
$1 < 2 ($not true$)$
$P(2): 2^2 < 2^2$
$4 < 4 ($not true$)$
$P(3): 3^2 < 2^3$
$9 < 8 ($not true$)$
$P(4): 4^2 < 2^4$
$16 < 16 ($not true$)$
$P(5): 5^2 < 2^5$
$25 < 32 ($true$)$
$P(6): 6^2 < 2^6$
$26 < 64 ($true$)$
Thus, $n^2 < 2^n$ for all natural numbers $n ≥ 5.$

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