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.

Mathematical induction is a form of deductive reasoning. Its most common use is induction over well-ordered sets, such as natural numbers or ordinals. While induction can be expanded to class relations which are well-founded, this tag is aimed mostly at questions about induction over natural numbers.

In general use, induction means inference from the particular to the general. This is used in terms such as inductive reasoning, which involves making an inference about the unknown based on some known sample. Mathematical induction is not true induction in this sense, but is rather a form of proof.

Induction over the natural numbers generally proceeds with a base case and an inductive step:

  • First prove the statement for the base case, which is usually $n=0$ or $n=1$.
  • Next, assume that the statement is true for an input $n$, and prove that it is true for the input $n+1$.

The following variant goes without a base case: Assuming the statement is true for all $n\in\mathbb N$ with $n < N$, prove that is true for $N$, too. This has to be done for all $N\in\mathbb N$.

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Group of square free order with a normal $p$-Sylow is solvable

Let $G$ be a group of order $p_1...p_s$ where $p_1,...,p_s$ are distinct primes. If $G$ has a normal $p$-Sylow subgroup, then $G$ is solvable. We proceed by induction on $s$. If $s = 1$, $G$ is cyclic, so $G$ is abelian and therefore it is…
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There exists $i, j \in \mathbb{N}$ such that $n=3i+5j$ for $n\ge 8$

Prove that there exists $i, j \in \mathbb{N}$ such that $n=3i+5j$ for $n\ge 8$ I'm having a hard time with this exercise, I'm trying to prove it by induction: Basis step: $n=8 \implies 8=3\cdot1+5\cdot 1$ $n=9 \implies 9=3\cdot3+5\cdot0$ $n=10…
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Strong Induction for Fibonacci

I'm a little lost of how to use strong induction to prove the following for the Fibonacci sequence: $F_n < 2^n$ for all natural numbers Any help would be very much appreciated!!
Robin
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Proof by Induction of a Recursively Defined Sequence

The question I'm currently struggling with is: Consider the following recursively defined sequence: $$\ a_1 = 0, a_{n+1}= \frac{a^2_n+4}{5} $$ a) Show by induction that $0\le a_n \le 1 $ for all n b) Prove that if $a_n$ converges, so that…
king
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Formula for this sequence?

$P_n= \prod^n_k_=_2 \frac{k^2-1}{k^2}$ for $n \ge 2$ I calculated $P_1 to P_3$ . I have been trying to come up with a formula but I can't really see any pattern. $P_2 = \frac{3}{4} , P_3 = \frac{2}{3}, P_4 = \frac{5}{8}$
user60334
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Why restrict to $\Sigma_1^0$ formulas in $RCA_0$ induction?

I'm reading Stillwell's Reverse Mathematics, and the induction axiom was just introduced. For a $\Sigma_1^0$ formula $\phi$, \begin{equation} [\phi(0)\; \wedge\; \forall n\, (\phi(n) \Rightarrow \phi(n+1))] \Rightarrow \forall n\,\phi(n) …
luqui
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Proving the connection between the Binomial Theorem and the product rule for derivatives

Let $a(x)$ and $b(x)$ be smooth functions, i.e they are infinitely times differentiable. I have made the assumption that the derivative for the function $$f(x)= (a\cdot b)(x)$$ can be given by…
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Validity of Using Induction to Show Union of an Infinite Ascending Chain of Subgroups is a Subgroup

Can this be done by induction instead of just proving the subgroup criterion? I can prove using the essentials tools of group theory, but looking at the problem, I was wondering if we can simply use an induction argument. Given we have a chain of…
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Proof by induction: $3^n|a_n\ $

$$Problem$$ Given $ (a_n)_{n\in N}$ the sequence defined by: $$a_1=15,\ a_2=18,\ a_{n+2}=6a_{n+1}-7a_n^4,\ \forall n\in \mathbb{N}$$ Prove by induction that $\forall n \in N$ a) $3^n|a_n$ b) $3^{n+1}\not| \ a_n $ I don't know how to face this…
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How to solve $\frac{1}{1*2} + \frac{1}{2*3} + \frac{1}{3*4} + \dots + \frac{1}{n(n+1)} = \frac{n}{n+1}$

I am stuck on factoring out everything properly. I feel like I am combining these fractions wrong or something because I always have an extra 1. edit: edit: I am still stuck. Math isn't working out, I am making a mess with the constant edits, I will…
Evan Kim
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How to prove $\sum^n_{i=1} \frac{1}{i(i+1)} = \frac{n}{n+1}$?

How can I prove that $\sum^n_{i=1} \frac{1}{i(i+1)} = \frac{n}{n+1}$? I noticed that in the sum, the denominator has terms that cancel out, but I'm not sure how to take advantage of that.
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Why doesn't one see more *induction on the number of primes* arguments?

I've used this proof technique to examine what happens when $ax = by$ for example. The proof worked out nicely by introducing the $n$th prime $p$. Here's exactly where I used it before What I want from you is either an example where such prime…
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Proof by induction help. I seem to be stuck and my algebra is a little rusty

Stuck on a homework question with mathematical induction, I just need some help factoring and am getting stuck. $\displaystyle \sum_{1 \le j \le n} j^3 = \left[\frac{k(k+1)}{2}\right]^2$ The induction part is: $\displaystyle…
Max
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Show that if $2^k\in S\space\space\forall k\in\mathbb{N}$, and if $k\in S$ and $k\ge2$, then $k-1\in S$, then $S=\mathbb{N}$

Show that if $2^k\in S\space\space\forall k\in\mathbb{N}$, and if $k\in S$ and $k\ge2$, then $k-1\in S$, then $S=\mathbb{N}$ My attempt: Let $n\in\mathbb{N},$ hence $2^n\in\mathbb{N}$ by the first assumption. Now denote the set $A$ by the following:…
user573025
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Confused about a mathematical induction detail

Suppose I wish to prove that $\forall n\in\mathbb{N}\ge 5,$ the statement $\mathscr{P}(n): n^2<2^n$ is true. Proof: Proceeding by induction on $n$, we note that $\mathscr{P}(5):(5)^2=25<2^5=32 \implies \mathscr{P}(5) $ holds. Now we induct on $n$…
user573025
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