# 2 to the n is Greater than n Cubed when n is 10 and above

Jump to navigation
Jump to search

## Contents

## Theorem

- $\forall n \in \Z, n \ge 10: 2^n > n^3$

## Proof

Proof by induction:

For all $n \in \Z$ such that $n \ge 10$, let $P \left({n}\right)$ be the proposition:

- $2^n > n^3$

We note that:

- $2^9 = 512 < 729 = 9^3$

so when $n < 10$ the proposition does not hold.

### Basis for the Induction

$P \left({10}\right)$ is the case:

- $2^{10} = 1024 > 1000 = 10^3$

so $P \left({10}\right)$ is seen to hold.

This is our basis for the induction.

### Induction Hypothesis

Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 10$, then it logically follows that $P \left({k + 1}\right)$ is true.

So this is our induction hypothesis:

- $2^k > k^3$

We need to show that:

- $2^{k + 1} > \left({k + 1}\right)^3$

### Induction Step

This is our induction step:

We note that when $k \ge 10$:

\(\displaystyle \left({1 + \dfrac 1 k}\right)^3\) | \(=\) | \(\displaystyle 1 + \frac 3 k + \frac 3 {k^2} + \frac 1 {k^3}\) | Binomial Theorem | ||||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle 1 + \frac 3 {10} + \frac 3 {100} + \frac 1 {1000}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1 + \frac {300 + 30 + 1} {1000}\) | |||||||||||

\(\text {(1)}: \quad\) | \(\displaystyle \) | \(<\) | \(\displaystyle 2\) |

Thus:

\(\displaystyle 2^{k + 1}\) | \(=\) | \(\displaystyle 2 \times 2^k\) | |||||||||||

\(\displaystyle \) | \(>\) | \(\displaystyle \left({1 + \dfrac 1 k}\right)^3 2^k\) | from $(1)$ | ||||||||||

\(\displaystyle \) | \(>\) | \(\displaystyle \left({1 + \dfrac 1 k}\right)^3 k^3\) | Induction Hypothesis | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle k^3 + \frac {3 k^3} k + \frac {3 k^3} {k^2} + \frac {k^3} {k^3}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle k^3 + 3 k^2 + 3 k + 1\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \left({k + 1}\right)^3\) |

So $P \left({k}\right) \implies P \left({k + 1}\right)$ and the result follows by the Principle of Mathematical Induction.

$\blacksquare$

## Sources

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.1$: Mathematical Induction: Exercise $10$