# 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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### Difference between set theory proof and logic proof of complete induction

Set theory proof: Let $\mathbf{A}$ be the set such that $\{0,1,2,...,n\} \subset \mathbf{A} \implies n+1 \in \mathbf{A}$. Our goal is to show that $\mathbf{A} = \mathbb{N}$. To do this, we construct ...
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### How to prove if this $\sum_{l=0}^{n}\binom{n}{l}=2^{n}$ is valid for all $n\in \mathbb{N}$? [duplicate]

Prove for for all $n\in \mathbb{N}$: $\sum_{l=0}^{n}\binom{n}{l}=2^{n}$ I know the steps of induction but i have no idea how to prove this equation with binomial coefficient. 1) For the induction ...
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### Mathematical Induction Inequality problem [on hold]

I am trying to solve the following problem with mathematical induction: $$\forall n>1,\qquad \frac{1}{2^2}+\frac{1}{3^2}+\ldots+\frac{1}{n^2}<\frac{n-1}{n}$$ but since I am new to the concept ...
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### How can you prove that $1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$ without using induction?

Using mathematical induction, I have proved that $$1+ 5+ 9 + \cdots +(4n-3) = 2n^{2} - n$$ for every integer $n > 0$. I would like to know if there is another way of proving this result ...
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### How does the induction proof work in this solution?

Refer to answer 1.1 of this file: http://www.dei.unipd.it/~geppo/AA/DOCS/NPC.pdf From my understanding and this thread, http://math.stackexchange.com/a/928412, we need 3 steps for that proof. ...
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### Prove that $\sum_{i=1}^{n^2} \left \lfloor \sqrt{i} \right \rfloor = \frac{n(4n^2 - 3n + 5)}{6}$ using induction?

Clearly, it's true for n=1. Assuming true for n=k, we have $$\left \lfloor \sqrt{1} \right \rfloor + \left \lfloor \sqrt{2} \right \rfloor ..... + k = \frac{k(4k^2 - 3k + 5)}{6}$$ But how can we ...
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### Prove that if a collection of subsets of {1,..,n} that each pair of subsets has at least one element in common, there are at most $2^{n-1}$ subsets

Full question: Prove that if a collection of subsets of {1,2,...,n} has the property that each pair of subsets has at least one element in common, then there are at most $2^{n-1}$ subsets in the ...
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### Proof by induction: inequality $n! > n^3$ for $n > 5$

I'm given a inequality as such: $n! > n^3$ Where n > 5, I've done this so far: BC: n = 6, 6! > 720 (Works) IH: let n = k, we have that: $k! > k^3$ IS: try n = k+1, (I'm told to only work ...
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### Prove by mathematical induction that: $\forall n \in \mathbb{N}: 3^{n} > n^{3}$

Prove by mathematical induction that: $$\forall n \in \mathbb{N}: 3^{n} > n^{3}$$ Step 1: Show that the statement is true for $n = 1$: $$3^{1} > 1^{3} \Rightarrow 3 > 1$$ Step 2: Show ...
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### Prove by induction that $\det(A^T) = \det (A)$ [closed]

If $A$ is an $n\times n$ matrix then $\det(A^T) = det(A)$. Prove by induction that the matrix obtained by deleting the $i^{\rm th}$ row and $j^{\rm th}$ column of $A^T$ is the transpose of the ...
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### Proof by induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$

Prove via induction that $\sum_{i=1}^n \frac{i}{(i+1)!}=1- \frac{1}{(n+1)!}$ Having a very difficult time with this proof, have done pages of work but I keep ending up with 1/(k+2). Not sure when to ...
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