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Prove that: $$(n+3)^2 \le 2^{n+3},\quad n\in\mathbb{N}$$

Please show me how to prove this inequality using a method other than mathematical induction. I was solving some questions based on the principle of mathematical induction and after solving nearly $20$ questions, I noticed that there is always an alternate proof for a statement which I proved by using principle of mathematical induction. So, I was trying to prove the statement given in the title by an alternate method. I found that the inequality can be easily verified from the graph, but I wanted to know if it can be proved in any algebraic way without using graphs.

Some Guy
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Sameer Nilkhan
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    Here's a MathJax tutorial :) – Shaun Jan 02 '20 at 09:38
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    You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an [edit]): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. – Shaun Jan 02 '20 at 09:39
  • Inequality is not true for n=0. – Alessio K Jan 02 '20 at 09:39
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    @Extended not everyone includes $0$ in the natural numbers so that might not be a counterexample – postmortes Jan 02 '20 at 09:40
  • related: https://math.stackexchange.com/questions/517555/fastest-way-to-check-if-xy-yx/ – Henry Jan 02 '20 at 09:48
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    It is time that you started making an effort to format your questions. Math needs to be between $ symbols. Exponents of more than one character need to be enclosed in { }. I have just fixed your title - please look at it to see what I did. – almagest Jan 02 '20 at 09:50
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    Incidentally, why do you want to prove the inequality without using induction? – almagest Jan 02 '20 at 09:52

1 Answers1

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Look at $(x+3)\ln 2-2\ln (x+3)$. It is increasing on $[1,\infty)$. Now can you finish?

Kavi Rama Murthy
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