# Tagged Questions

For questions about mathematical induction, a method of mathematical proof. Mathematical induction generally proceeds by proving a statement for some integer, called the *base case*, and then proving that if it holds for one integer then it holds for the next integer. This tag is primarily meant ...

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### Using induction to prove a sequence is always less than a given number

Let $f(1)=2$ and $f(n+1)=\sqrt{3+f(n)}$. Prove that $f(n)<2.4$ for all $n\ge 1$. I established a base case when $n=1$ and then moved on to the inductive step by assuming the statement is true for ...
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### How to use Induction properly?

I would like to prove the following equation using induction. However that seems somehow impossible at least for me: $\sum\limits_{k=1}^{2n} {(-1)^k \cdot k^2}=(2n+1)\cdot n$ I tried to show that ...
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### How to prove through induction

How can I prove by induction that $$\binom{2n}n<4^n\;?$$ I have solved for the base case, $n=1$, and have formulated the induction hypothesis. I was thinking about Pascal's identity for the rest,...
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### Showing $1+2+\cdots+n=\frac{n(n+1)}{2}$ by induction (stuck on inductive step)

This is from this website: Use mathematical induction to prove that $$1 + 2 + 3 +\cdots+ n = \frac{n (n + 1)}{2}$$ for all positive integers $n$. Solution to Problem 1: Let the ...
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### question on prove by induction that for each n$\in\mathbb{N}_{\ge2}$, $n^2$< $n^3$

I have to prove by induction that for each n$\in\mathbb{N}_{\ge2}$, $n^2$< $n^3$. If I try to prove for P(1) I end up with 1 < 1. Is this right? Why does it or does it not make sense?
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### Is there a much simpler proof for Euler factorial formula?

Euler formula for factorial stated as follows Theorem [Euler]: For any non negative integers $a$ and $n$ such that $a\geq n$ $$n!=\sum_{k=0}^{n}(-1)^k\binom{n}{k}(a-k)^n$$ Proving this ...