Induction in general is the inference from the particular to the general. Mathematical induction is not true induction, but 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 ...

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Use induction on $n$ to prove that $2n+1<2^n$ for all integers $n≥3$.

Use induction on n to prove that $2n+1<2^n$ for all integers $n\geq 3$. My attempt: Let $P(n)$ be the statement $2n+1<2^n$. Base case: Prove that $P(3)$ is true. $LS = 2(3)+1=7$ and ...
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Induction problem: “if the group has at least one player who is better than Messi, then all the members of the group are better than Messi”

I'm having some trouble with the following problem: "A french man is trying to prove that any non empty group of french soccer players satisfies the following: 'if the group has at least one player ...
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34 views

Is it obvious that $\sum_n x_n = 0 $ when $x_n = 0 ~ \forall n \in \mathbb N$?

Until recently I used to think that because of induction, a statement $P_n$ which is true $\forall n \in \mathbb N$ was also true when $n \to \infty$. Life was simple, and I was happy. Then someone ...
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Got stuck in proving monotonic increasing of a recurrence sequence

I'm stuck with the prove of the following recurrence sequence which was part of and old exam. $a_1:=\frac{1}{2}, a_{n+1}:=a_n(2-a_n)$ for $n \in \mathbb{N}$ I have to show that $0 <a_n < 1$ ...
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Proof by Induction - Sequence of integers

Suppose a sequence of integers $a_1$, $a_2$, ... is defined as: $$a_1 = 3$$ $$a_2 = 6$$ $$a_n = 5a_{n-1} - 6a_{n-2} + 2$$ for all $n\ge3$ $\mathbf {Prove}$ $\mathbf {S(n)}$: $a_n = 1 + 2^{n-1} + ...
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proof by strong induction of single variable with exponent

$x^n + \frac{1}{x^n} \in \mathbb{Z}$ (is an integer), for all positive integers $n$, where $x$ is rational. I've surmised that the only rational numbers that satisfy $x$ are 1 and -1. Thus, as you ...
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43 views

Prove that $n^2 < 2^n$ for all $n \geq 6$

My approach to solving this: By induction. (1) $S(n) = (n^2 < 2^n)$ for all $n \geq 6$, $n \in \mathbb N$. (2) Base Case: $n = 6$ $$6^2 < 2^6$$ $$36 < 64$$ So the statement is true for ...
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Strong Induction: Finding the Inductive Hypothesis

Consider this claim: Every positive integer greater than 29 can be written as a sum of a non-negative multiple of 8 and a non-negative multiple of 5. Assume you are in the inductive step and trying ...
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Induction proof help

Let $x_1=1$ and $x_{n+1}=\frac{1}{x_n+1}$. Prove that for all $n \in \mathbb N$ $x_n >0$. So, $x_1=1>0$. Suppose $x_n>0$. I need to show that $x_{n+1}=\frac{1}{x_n+1}>0$. Isn't it ...
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60 views

Permuation, disjoint cycles proof by induction.

I am having a hard time writing out a general proof. Can anyone please help? Thank you. Exercise: Show that any k-cycle (a1,......,ak) can be written as a product of some number of (k-1) 2-cycles. ...
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89 views

Properties of Natural Numbers and Mathematical Induction

When working with natural numbers how to check that the property we consider is "permissible" to speak about? And not like the property "The smallest positive integer not definable in under eleven ...
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97 views

What is mutual induction and how does that differ from regular induction?

http://web.cecs.pdx.edu/~black/CS311/proof_by_mutual_induction.pdf I read this and I fail to see any difference. It's the same thing, prove for n = 0 and then prove for n = k+1.
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If I'm asked to prove that $n \le m$, is it sufficient to show that $n < m$?

I have a homework question, which is to prove by induction that $\sum\limits_{r=1}^{n} \frac{1}{\sqrt{r}} \leq 2\sqrt{n}$ for every integer $n \geq 1$. I've managed to show by induction that ...
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187 views

prove $\binom{n}{k}\frac{1}{n^k}\leq\frac{1}{k!}$

i am learning maths so fast here in MSE, thank you guys so much for being here to help us! so now, my next step towards proficiency: :). i am trying to prove that ...
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107 views

How to prove $4(n!)>2^{n+2}$ for $ n\geq 4$ with induction

I've done the base step, but how do I prove it is true for $n+1$ without using a fallacy? $$4(n!)>2^{n+2}\quad \text{for } n\geq 4$$ Please help.
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1answer
49 views

Induction: $\frac{n!}{x(x+1)\cdots(x+n)} = \binom{n}{0}\frac{1}{x}-\binom{n}{1}\frac{1}{x+1}+\cdots+(-1)^n\binom{n}{n}\frac{1}{x+n}$

$$\frac{n!}{x(x+1)\cdots(x+n)} = \binom{n}{0}\frac{1}{x}-\binom{n}{1}\frac{1}{x+1}+\cdots+(-1)^n\binom{n}{n}\frac{1}{x+n}, \quad \text{for } x \not \in \{0,-1,-2,\dots,-n\}$$ Can somebody please help ...
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5answers
227 views

Induction: $n^{n+1} > (n+1)^n$ and $(n!)^2 \leq \left(\frac{(n + 1)(2n + 1)}{6}\right)^n$

How do I prove this by induction: $$\displaystyle n^{n+1} > (n+1)^n,\; \mbox{ for } n\geq 3$$ Thanks. What I'm doing is bunch of these induction problems for my first year math studies. I tried ...
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69 views

Induction proof of function from $\mathbb R$ to $\mathbb R$

Let f be a function from $\mathbb R$ to $\mathbb R$ satisfying $f(\frac{x_1+x_2}{2})=\frac{f(x_1)+f(x_2)}{2}$ Prove that for any positive integer $n$ we have ...
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55 views

Induction with multiple variables

Let the function g : R $\rightarrow$ R satisfy $g(xy) = x \cdot g(y) +y \cdot g(x)$ for all real numbers x and y. Prove $g(u^n) = nu^{n-1}g(u)$, for all positive integers $n$ and all real numbers ...
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151 views

How to prove this with induction

$$(P_0 \lor P_1 \lor P_2\lor\ldots\lor P_n) \rightarrow Q $$ is the same as $$(P_0 \rightarrow Q) \land (P_1 \rightarrow Q) \land (P_2 \rightarrow Q) \land\ldots\land(P_n \rightarrow Q)$$ Do I ...
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155 views

Proving strings [duplicate]

We consider strings of n characters, each character being a, b, c, or d, that contain an even number of as. (0 is even.) Let $E_n$ be the number of such strings.Prove that for any integer $n ...
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42 views

Prove GM-AM inequality using induction

Show that $G_{2^n}\le A_{2^n}$ by using induction on n. I've proven the base case in the previous exercise: Let $G_2=\sqrt{a_1a_2}$ and $A_2=\frac{1}{2}(a_1+a_2)$ and $a_1,a_2 \in \mathbb{R}$ ...
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3answers
436 views

Proof by Induction: Solving $1+3+5+\cdots+(2n-1)$

The question asks to verify that each equation is true for every positive integer n. The question is as follows: $$1+ 3 + 5 + \cdots + (2n - 1) = n^2$$ I have solved the base step which is where ...
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Induction of $A_i$ [duplicate]

The base case $n=1$: $B\cup\left(\bigcap_{i=1}^1A_i\right)=B\cup A_1$ and $\bigcap_{i=1}^1(B\cup A_i)=B\cup A_1$. Now, suppose inductively that ...
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Discrete Math: Ways to Prove Induction

The point of mathematical induction is to prove $\forall x\geq b[P(x)]$ by instead proving $P(b)\wedge \forall x\geq b[P(x)\rightarrow P(x+1)]$ ($b$ is often, but not always, $0$ or $1$). However, ...
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What's mutual induction and structural induction?

From what I've read, structural induction is when you have 2 or more statements you want to prove and you prove the second statement by proving the first statement beforehand and using it as a part of ...
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Proof that $n^2 < 2^n$

How do I prove the following statement by induction? $$n^2 \lt 2^n$$ $P(n)$ is the statement $n^2 \lt 2^n$ Claim: For all $n \gt k$, where $k$ is any integer, $P(n)$ (since $k$ is any integer, I ...
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PROVE if $x \ge-1 $then $ (1+x)^n \ge 1+nx $ , Every $n \ge 1$

Use mathematical induction to prove this. Here is my answer but I stuck at certain point. Base Case: n=1 $$(1+x)^1 \ge 1+x $$ True , Induction Case: n=k assume $$(1+x)^k ...
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Proving by induction propositions of the type $P(n_1, n_2, …, n_k)$, where $n_1, n_2, …,$ and $n_k$ are natural numbers

For example: I've seen proofs of the multinomial theorem that use induction in the number of terms that are elevated at some power, but none that use induction in the exponent instead of using it in ...
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Prove the following using induction on n (matrices)

Prove the following using induction on n: $$\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}^{n} = \begin{pmatrix} n+1 & n \\ -n & -n+1 \end{pmatrix}$$ I know that multiplication of ...
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274 views

Proof by induction involving fibonacci numbers

The Fibonacci numbers are defined as follows: $f_0=0$, $f_1=1$, and $f_n=f_{n−1}+f_{n−2}$ for $n≥2$. Prove each of the following three claims: (i) For each $n≥0$, $f_{3n}$ is even. (ii) For each ...
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Prove that the inequality $(1+ \frac{1}{n})^n < n$ holds for all $n \geq 3$

First we need to prove the basis. If we let $n=3$, then $(1+ \frac{1}{3})^3 < 3$ $(\frac{3}{3}+ \frac{1}{3})^3 < 3$ $(\frac{4}{3})^3 < 3$ $(\frac{64}{27}) < 3$ The inequality ...
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Show $0 \leq e^{-x} - \left( 1 - \frac{x}{n} \right)^n \leq \frac{x^2e^{-x}}{n}$ by induction

Show that if $0\leq x < n, n \geq 1$, and $n\in\mathbb{N}$ then $$ 0 \leq e^{-x} - \left( 1 - \frac{x}{n} \right)^n \leq \frac{x^2e^{-x}}{n}. $$ By using induction. Progress: Decided to split ...
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3answers
85 views

Prove by induction: If $h$ and $k$ are any two distinct integers, then $h^n-k^n$ is divisible by $h-k$.

If $h$ and $k$ are any two distinct integers, then $h^n-k^n$ is divisible by $h-k$. Let's start with the basis. Let $n=1$, then $h^1-k^1 = h-k$ Now for the induction, I can't use $k$ because I ...
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212 views

Strong Induction and Recursion

Consider the recursion given by \begin{equation}f(n) = 2f(n−1)− f(n−2)+6 \text{ for } n ≥ 2 \text{ with } f (0) = 2 \text{ and }f (1) = 4 \end{equation} Use mathematical induction to prove that ...
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Number of bitstrings with $000$ as substring

I have $F_n$ number of bitstrings that have $000$, How would I prove that for $n \ge 4$ , $a_n = a_{n-1} +a_{n-2}+a_{n-3}+ 2^{n-3}$? Now there are many ways to go about this but if I choose starting ...
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From $ P( \cup_{n=1}^m A_n ) \leq \sum_{n=1}^m P( A_n ) $ to $ P( \cup_n A_n ) \leq \sum_n P( A_n ) $?

Let $P$ be a probability measure and $(A_n)_n$ a sequence of events. I am trying to understand the proof of the following property: $$ P( \cup_n A_n ) \leq \sum_n P( A_n ) $$ The proof I read goes ...
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Tangent equation divisible by (x-y)

I have attempted this proof but I am not sure is the induction step is correct any assistance would be appreciated also I am not sure if i have proved what I was trying to. Let ...
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39 views

Splitting up bracket terms

I found a statement saying: Let $\circledast $ be an associative binary operation on a set $\mathbb{X}$. A bracket term of length n, consisting of n elements $a_1, ..., a_n$ and arbitrary brackets, ...
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443 views

Show that $e^x > 1 + x + x^2/2! + \cdots + x^k/k!$ for $n \geq 0$, $x > 0$ by induction

Show that if $n \geq 0$ and $x>0$, then $$ e^x > 1 + x + \frac{x^2}{2!} + \dots + \frac{x^n}{n!}.$$ Not sure where to get started with this induction proof.
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351 views

What is wrong with my induction proof?

The Fibonacci sequence is defined by $a_1 = 1, a_2 = 1$ and for all $n \ge 2, a_{n+1} = a_n + a_{n-1}$. Thus the sequence begins $$1,1,2,3,5,8,13,21,...$$ prove that for all $n \ge 1, a_n < ...
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113 views

Using induction, prove that $(-7)^n -9^n$ is divisible by $16$

First of all, I think the problem should be $(-7)^n -9^n$ is divisible by $-16$ because if I test the basis by letting $n=1$, I have $-16$ instead of $16$. Edit: Alright ... I sort of understand why ...
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96 views

Using induction to prove a general form from a recurrence relation

I have the recurrence relation: $a_{n+2} = \dfrac{-a_{n}}{(n+2)(n+1)}$. I have done some work to identify that two cases emerge: one is n is positive, and one if n is negative. If n = 2m (even) ...
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85 views

Using Induction, prove that $107^n-97^n$ is divisible by $10$

Using Induction, prove that $107^n-97^n$ is divisible by $10$ We need to prove the basis first, so let $ n = 1 $ $107^1-97^1$ $107-97 = 10$ This statement is clearly true when $ n = 1 $ Now ...
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1answer
38 views

Small question about inductive proof about rational sequences

I am writing an inductive proof about this: the description is not terribly important so you don't have to read that. here's my question: let $P(n)$ be the statement that $x_n$ is a rational ...
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56 views

How to show $((k+1)!)^2 2^k \leq (2(k+1))!$

How do you show that $((k+1)!)^2 2^{k+1} \leq (2(k+1))!$ This is part of an induction proof and I have not made any progress.
2
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2answers
124 views

show that $2n\choose n$ is divisible by 2 [duplicate]

I tried using induction, but in the inductive step, I get: If $2n\choose n$ is divisible then I want to see that $2n +2\choose n +1$ $${2n +2\choose n +1} = (2n + 2)!/(n+1)!(n +1)! = {2n\choose n} ...
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53 views

Induction Problem: $\sum_{j=1}^{n}(-1)^j\binom{n}{j}\frac{1}{j+1}=-\frac{n}{n+1}$

Prove that: $$\sum_{j=1}^{n}(-1)^j\binom{n}{j}\frac{1}{j+1}=-\frac{n}{n+1} $$ I know that 1.I need prove it by induction 2.this can be helpful : $ ...
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2answers
216 views

Is it possible to play the Tower of Hanoi with fewer than $2^n-1$ moves?

The Tower of Hanoi game consists of three identical upright pegs and n rings all of different diameters that can be stacked over any of the pegs. Initially, all of the rings are stacked around one of ...
0
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117 views

Two very difficult induction proofs; having trouble with the inductive step

$$\sum_{k=0}^{n-2} \binom{n}{k}\binom{n}{k+1}\frac{n-2k-1}{k+1} = n-2 + \frac{1}{n+1}\binom{2n}{n}$$ $$\sum_{k=0}^{n-2} \binom{n}{k}\binom{n}{k+2}\frac{n-2k-1}{k+1} = -n + ...