# 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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### Prove that the identity is true for all natural numbers [closed]

for the identity: $$\frac{n!}{x(x+1)(x+2)...(x+n)} = \frac{A_0}{x+0} + \frac{A_1}{x+1}+...+ \frac{A_n}{x+n}$$ prove $$A_k= (-1)^kC(n,k)$$ I think this might work by induction, but i am not able to ...
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### Expanding a Proof of Induction on $\Bbb N$ to $\Bbb Q$ (Linear Algebra)

My problem is the following: I have an $\Bbb R$ Vectorspace called $V$ and had to show via induction that $\langle nv, w \rangle=n \langle v, w \rangle$ for $v,w \in V$ and $n\in \Bbb N$. (it's not ...
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### Proof by induction, 1 · 1! + 2 · 2! + … + n · n! = (n + 1)! − 1

So I'm supposed to prove that $$1 · 1! + 2 · 2! + \dots + n · n! = (n + 1)! − 1$$ using induction. What I've done Basic Step: Let $n=1$, $$1\cdot1! = 1\cdot1 = 1 = (n+1)!-1 = 2!-1 = 2-1 = 1$$ ...
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### Discrete math induction proof (divisibilty) [duplicate]

How to show that $10^n -(-1)^n$ is always divisible by $11$ through proof of induction?
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I'm trying to prove $$\sum_{j=1}^{n-1} jx^j = \frac{x-nx^n+(n-1)x^{n+1}}{1-x^2}$$ holds for all positive integers $n$ and real $x\ne \pm 1$ by induction. In the inductive step I get \begin{align}\... 1answer 22 views ### Infimum and supremum of finite ordered subsets I am currently taking an introductory proofs course, and I have come across this problem. It's asking to prove the following: Let S be an ordered set. Let A be a non-empty finite subset. Then ... 2answers 60 views ### How can I prove by induction that this is a closed form of the Fibonacci sequence? [duplicate] How can I prove by induction that this is a closed form of the Fibonacci sequence?F_n=\frac1{\sqrt5}\left(\frac{1+\sqrt5}2\right)^{n+1}-\frac1{\sqrt5}\left(\frac{1-\sqrt5}2\right)^{n+1} I've ...
Proof by induction that the coefficients of $(a+b)^n$ in order, if place as a number, the first coefficient being having the biggest place value, and each number lowers in place value, are equal to ...
Let $H_n = \sum_{j=1}^n \frac 1j$. I'm trying to prove that $H_{2^n}\ge 1+\frac n2$ using the principle of mathematical induction. The base step was no problem but I'm stuck on the inductive step: \$...