# 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 by induction $3+3 \cdot 5+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$

My question is: Prove by induction that $$3+3 \cdot 5+ 3 \cdot 5^2+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$$ whenever $n$ is a nonnegative integer. I'm stuck at the basis step. If I ...
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### How to get a feel for rigor/form used in mathematics?

I'm an engineer, and while you get introduced to many concepts of mathematics, but only with a subset of the vocabulary, and none of the rigor and proofs. So while trying to read a mathematical book, ...
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### Inequality : $\displaystyle \sum_{k=1}^n x_k\cdot \displaystyle \sum_{k=1}^n \dfrac{1}{x_k} \geq n^2$

I have to show the inequality of $$\left(\sum_{i=1}^n x_i\right)*\left(\sum_{i=1}^n \frac{1}{x_i}\right) \geq n^2.$$For $x_1, ... x_n \in \mathbb{R_{>0}}$ and $n \geq 1$. I wanted to show this ...
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### Well-ordering principle and theorem

Could somebody clearly explain the difference between the well-ordering principle and the well-ordering theorem? Is one of these related to the Principle of Mathematical Induction, and the other to ...
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### Fibonacci Sequence: Prove $f_1+f_3+\dots+f_{2n-1}=f_{2n}$ by Induction.

I believe the majority of my proof is correct I'm just not certain about the base case if any one can explain how to do that base case or fix any error I made I would greatly appreciate it. Recall ...
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### induction proof over graphs

I have a question about how to apply induction proofs over a graph. Let's see for example if I have the following theorem: Proof by induction that if T has n vertices then it has n-1 edges. So what ...
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### trouble undestanding the proof for the therom “If x is element of N and x != 1, then there is a unique y so that x = y'.”

give the following axioms The following theorem is proven Im having trouble understanding the sentence from "if x=1 then x' element of N ..." up to "and by definition of A, x' element of A." ...
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### Strong Induction issue

I am trying to prove a statement using strong induction but I seem to be getting stuck. I don't know if did something wrong or I am just not recognizing an opportunity for factoring/how to factor ...
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### prove by mathematical induction $1^{3}+2^{3}+…+n^{3}=(n(n+1)/2)^{2}$ [duplicate]

I already done the basis step or prove of one p(1). From this point,this is my hypothesis: $k^{3}$=$(k(k+1+1)/2)^{2}$ I wish to prove that my hypothesis is equal to $(k+1)^{3}$=$(k+1(k+1+1)/2)^{2}$ ...
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### Show the sum of the squares of the first $n$ positive integers is $[n(n+1)(2n+1)]/6$ for all $n$ greater than or equal to $2$ [duplicate]

I need to show by proof that the statement: The sum of the squares of the first n positive integers is $[n(n+1)(2n+1)]/6$ for all n greater than or equal to $2$ is true. I know im going to ...
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### Why do you need to show A(1) before proving A(n) by induction? [duplicate]

My instructor stated that in order to have a valid proof by mathematical induction, you first have to show A(1) holds, and then assume A(n) to deduce A(n+2). Why is the first step necessary if we are ...
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### Proof by math induction with inequality example, why is “replacement” allowed?

I have trouble with the understanding of mathematical induction concerning inequalities. For example, the question is: Prove by mathematical induction that $n ^ 2 <2 ^ n$ if $\forall n \in {N}$ ...
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### Prove that reverse of regular L is also regular [duplicate]

Prove that reverse of regular language is also regular. I know, how i can to this by using DFA of L. Changing directions of edges and so on. But how can it be done with Structural induction? What ...
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### Is this type of proof by induction correct in Sylow's Theorem?

The following is the first part of the Sylow's Thm: My question is: if order of $G$ was $p^a$ (and not $p^am$) then we could start with $|G|=1$ which means $a=0$. Then supposed that the theorem ...
### Prove by induction the particular inequality $\left(1.3\right)^n \ge 1 + \left(0.3\right)n$ for every $n \in \mathbb N$
$\left(1.3\right)^n \ge 1 + \left(0.3\right)n$ for every $n \in \mathbb N$ Not sure where I'm going wrong in my Algebra, but I assume it's because I'm adding an extra term. Is the extra term ...