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I'm trying to do homework problems and for the most part I've been getting the results. For this one though, I am having some trouble since its $2^n$ and I can't relate it properly:

Prove using simple induction that $n < 2^n$ for all $n \ge 1$.

So obviously, the basis step holds as $1 < 3$. Now, I assume that $n = k$ holds as well and have to prove for $n=k+1$. This is the step I am having trouble with as I cannot relate the induction hypothesis with what I want to end up with. Can anybody show me a model solution for this one? I think my trouble comes because of the $2^n $

Caleb Stanford
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rachxau
  • 29

4 Answers4

3

For the inductive step:

$$n+1\le 2n<2\times 2^n=2^{n+1}$$

3

a hint: you need to show that $$ n + 1 < 2^{n+1}$$ observe that $$ 2^{n+1} = 2 \cdot 2^{n} = 2^n + 2^n$$ now use your induction hypothesis to estimate $2^n$. also observe that $1$ is the smallest natural number, i.e. $1 \leq n$

mm-aops
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0

Hints:

  1. For all $n\ge 1$, $2^n>n>1$
  2. $2^{n+1}=2\cdot 2^n=2^n+2^n>2^n+1$
vadim123
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Yo know that $k<2^k$ and you want to prove $k+1<2^{k+1}$, but it's equal:

$$k+1<2^{k+1}=2^k+2^k$$

By induction hipothesis you know that $k<2^k$ and on the other hand it's clear that $1 \leq 2^{k}$. If you add this inqeualities side by side you get $k+1<2^k+2^k=2^{k+1}$.

agha
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