Prove for what natural numbers n the inequality $2^{n}$ > 2n + 1 is valid ?

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Solution

The inequality obviously does not hold for n = 1 and 2.
But the inequality holds for n = 3, for we have
2 $^{3}$ > 2.3 + 1
Now assume
2 $^{m}$ > 2m + 1
where m is a natural number > 3.
But 2 $^{m}$ > 2 for all m > 1.
Adding the two inequalities we obtain for m > 3.
2 $^{m}$ + 2 $^{m}$ > 2m + 1 + 2.
that is , 2 $^{m + 1}$ > 2(m + 1) + 1.
Hence the inequality holds for n = m + 1.
It follows by mathematical induction that the inequality
2 $^{n}$ > 2n + 1 holds for all natural numbers n $\geq$ 3

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Prove for what natural numbers n the inequality 2n > 2n + 1 is valid ?

Prove for what natural numbers n the inequality $2^{n}$ > 2n + 1 is valid ?