# Questions tagged [induction]

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 for questions about induction over natural numbers but is also appropriate for other kinds of induction such as transfinite, structural, double, backwards, etc.

7,006 questions
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### Proving the number of internal nodes of a binary tree

Let $T$ be a nonempty 2-3 tree, so that it includes at least one node. Prove that if $T$ represents a subset $S\subseteq E$ such that $|S|= n\in \Bbb N$, then $T$ has at most $n-1$ internal nodes. I ...
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### Strong induction

I have this question, Prove, $7 + 77 + 777 +7777 + 77...$n digits..$77 = 7/81[(10^n × 10) - 9n - 10]$ By induction. Now since this question was given in the exercise that involves proving various ...
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### Two dimensional induction

I have the following problem: I need to prove that given the following integral $\int_{0}^{1}{c(k,l)x^k(1-x)^l}dx = 1$, we the constant $c(k,l) = (k+l+1) {{k+l}\choose{k}} = \frac{(k+l+1)!}{k!l!}$, ...
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### Proof of Binomial-coefficients sum [duplicate]

How could I show by induction that this sum is true? $${n \choose 0}^2+{n \choose 1}^2+{n \choose 2}^2+...+{n \choose n}^2 = {2n \choose n}$$ All help is appreciated!
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### Property takagi function by induction.

In a lecture on Applied Functional Analysis, the professor showed us some properties of the Takagi function from this paper. He wrote at the end the following property and said it could be easily done ...
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### Proving the first principle of mathematical induction

I was asked to prove the first principle of mathematical induction without using the well ordered principle. If someone can elucidate the steps clearly it would be a great help ! Thank you!
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### What is a good example of prefix induction?

the Wikipedia article for Mathematical induction introduces a few variations of the classic principle, such as the strong induction. The strong induction comes with a few examples, namely the closed ...
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### How to show that $(n!/n^n)_{n\geq 0}$ is a null sequence?

How to show that $\left(\frac{n!}{n^n}\right)_{n\geq 0}$ is a null sequence, meaning it tends to zero. i tried proving that $n^n\geq n!$ for $n\in \mathbb{N}:n>0$. Is that a correct method? Base ...
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### How to prove that $n < n! - 1$ for $n > 2.$? [closed]

How to prove that $n < n! - 1$ for $n > 2.$? I have tried it by induction but I got stucked in the induction step in proving $n +1< (n + 1)! - 1$ for $n + 1> 2$. Could anyone help ...
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### Inductive Proof of Group with Prime Decomposition is Isomorphic to Direct Product of Cyclic Groups

My lecturer set as a bonus exercise the following induction proof: If $G$ is a finite abelian group $|G| = p_1^{n_1} \cdots p_s^{n_s}$ is the decomposition of $|G|$ into a product of distinct prime ...
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### Let $P(n)$ be : the sum of the first odd natural numbers equal $n^2$. Express in summation notation and use induction.

I presented the first odd $n$ natural integers as $2n+1=n^2$. In summation notation I just factored this into $(n-1)(n-1)$. I am unsure whether this is correct and also I don't know how to carry out ...
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### Let $a$ be a natural number not divisible by $5$

Let $a$ be a natural number not divisible by $5$ Prove that $8a^{8n}+a^{4n}-4$ is a multiple of $5$ for all n natural number. My try : Using induction Let : $A=8a^{8n}+a^{4n}-4$ For n=0 then ...
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### Self-complementary graph with 4k+1 vertices, which is 2k-regular [duplicate]

Prove that ∀k∈N, k≥1, there is a self-complementary graph with 4k+1 vertices, which is 2k-regular. I think that the best way to prove it is by induction. Any helpful suggestions? (I know that this ...
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### Question about self-complementary graphs

Question Prove that for every integer k≥1, exists a self-complementary graph with 4k vertices half of which are of degree 2k-1 and the other half of degree 2k. My approach So, I think the easiest way ...
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### proof by mathematical induction (n)!< (n)^n

"Let P(n) be the statement that (n)! < (n)^n, where is an integer greater than 1. Prove by mathematical induction that P(n) is true for all integers n greater than 1." I've written Basic step ...
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