Unable to solve an Induction Problem......

Prove that all integers $n>9$ satisfy the inequality $2^n>n^3$ .

#NumberTheory

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

For any positive integer $\displaystyle k$ , we have $\displaystyle \dfrac{k+2}{k+1} < \dfrac{k+1}{k}$

$\displaystyle {2}^{\frac{1}{3}} > \dfrac{k+1}{k}$ if $\displaystyle k>9$ by observation.

Thus, for all integers $\displaystyle n>9$ , we have, cubing both sides $\displaystyle 2k^3>(k+1)^3$ . It'll help us afterwards.

Canceling $\displaystyle k^3$ from both sides, we have $\displaystyle k^3>3k^2+3k^1$

By observation, for $\displaystyle k=10$ , the base case, it is true that $\displaystyle 2^k>k^3$ .

Let it be true for any $\displaystyle k \geq 10$ . Then we have $\displaystyle 2^k>k^3>3k^2+3k+1$ . Thus, $\displaystyle 2^k+k^3>k^3+3k^2+3k+1=(k+1)^3$ .

Since $\displaystyle 2^k>k^3$ , we can replace $\displaystyle k^3$ by $\displaystyle 2^k$ .

Finally getting $\displaystyle 2^{k+1}>(k+1)^3$ .

Hence, we have proved that if the claim is true for any integer $\displaystyle k>9$ , it is also true for $\displaystyle k+1$ .

Hence, the claim is true for all integers $\displaystyle n>9$

Satvik Golechha - 6 years, 4 months ago

What is the proof for your second step ? You said it is by observation. Is there any more complete way of proving the statement true for k+1, ie proving the inequality 2^(k+1)>(k+1)^3 from the induction hypothesis 2^k>k^3 for some k>9 ?

Venkata Karthik Bandaru - 6 years, 4 months ago

That's what I did, but in the opposite order. Try using the facts I mentioned to get it in the right order.

Satvik Golechha - 6 years, 4 months ago

@Satvik Golechha – Satvik, are you preparing for RMO ? If so, please help me bro !

Venkata Karthik Bandaru - 6 years, 4 months ago

@Venkata Karthik Bandaru – I just gave RMO in 10th, and got 5 outta 6 correct, but still I didn't get selected. If you want a book, try "An Excursion in Mathematics".

Satvik Golechha - 6 years, 4 months ago

@Satvik Golechha – Oh my god, you mean to say that one needs to get a full score ? I am in class 9 presently, and after seeing previous year RMO papers, @-@ GONE MAD .Thanks for your suggestion, I will try that book.

Venkata Karthik Bandaru - 6 years, 4 months ago

@Venkata Karthik Bandaru – I never say that you need a full score. It's just that I solved 5 of 6, and didn't get selected. Maybe they didn't like my methods or solutions. And moreover, the papers of the recent years are a bit easier than those of the past years. All the best!

Satvik Golechha - 6 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$