Prove the assertions of the following problems
Prove that $2^{n} > n^{2}$ for any natural number $n > 5.$

Open in App

Solution

$n > 5$
$⟹ (n - 1)^{2} > 2$
$⟹ n^{2} - 2 n - 1 > 0$
$⟹ 2 n^{2} - 2 n - 1 > n^{2}$
$⟹ 2 n^{2} > n^{2} + 2 n + 1$
$⟹ 2 n^{2} > (n + 1)^{2}$
Let us assume $2^{n} > n^{2}$
$2^{n + 1} = {2.2}^{n} > 2 n^{2} > (n + 1)^{2}$ which is true
By mathematical induction
$2^{n} > n^{2}, w h e n (n) > 5$

Suggest Corrections

Similar questions

Q. Prove the assertions of the following problems
Prove that the expression

n^{3} - n

is divisible by

24

for any odd

n

Q. Problems Set at an Oral Examination.
Prove that the following equations hold for any natural n.

(1 - \frac{1}{4}) (1 - \frac{1}{9}) (1 - \frac{1}{16}) . . . (1 - \frac{1}{n^{2}}) = \frac{n + 1}{2 n} .

Prove that $(1 - \frac{1}{2^{2}}) (1 - \frac{1}{3^{2}}) (1 - \frac{1}{4^{2}}) . . . . . . (1 - \frac{1}{n^{2}}) = \frac{n + 1}{2 n}$ for all natural numbers, $n \geq 2$ .

Q. Prove that the following equations hold for any natural n.
Prove that

(C_{n}^{o})^{2} + (C_{n}^{1})^{2} + . . . + (C_{n}^{n})^{2} = C_{2 n}^{n} .

7^{2 n} + 2^{3 n - 3} {.3}^{n - 1}

is divisible by

25

for any natural number

n \geq 1

. Prove that by mathematical

Join BYJU'S Learning Program

Prove the assertions of the following problems Prove that 2n>n2 for any natural number n>5.

n>5⟹(n−1)2>2⟹n2−2n−1>0⟹2n2−2n−1>n2⟹2n2>n2+2n+1⟹2n2>(n+1)2Let us assume 2n>n22n+1=2.2n>2n2>(n+1)2 which is true By mathematical induction 2n>n2,when (n)>5

Prove the assertions of the following problems
Prove that $2^{n} > n^{2}$ for any natural number $n > 5.$

$n > 5$
$⟹ (n - 1)^{2} > 2$
$⟹ n^{2} - 2 n - 1 > 0$
$⟹ 2 n^{2} - 2 n - 1 > n^{2}$
$⟹ 2 n^{2} > n^{2} + 2 n + 1$
$⟹ 2 n^{2} > (n + 1)^{2}$
Let us assume $2^{n} > n^{2}$
$2^{n + 1} = {2.2}^{n} > 2 n^{2} > (n + 1)^{2}$ which is true
By mathematical induction
$2^{n} > n^{2}, w h e n (n) > 5$