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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Understanding an application of induction in a proof

I understand what is done below, however I don't understand the induction used, especially not when applying $0\leq n$. If $s_n+1=f(s_n)$ with $|f′(x)|\leq 1/2$ prove that the sequence $s_n$ ...
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3answers
76 views

Use mathematical induction to prove a statement [on hold]

Use mathematical induction to prove that: $$A\cap\left(\bigcup_{i=1}^nB_i\right) = \bigcup_{i=1}^n\left(A\cap B_i\right)$$
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2answers
19 views

Help with solving mathematical induction problem

I need help with the following: Use mathematical induction to prove that for every $n\in N$, $$ \sum_{k=1}^n\frac{1}{\cos kx \cos(k+1)x}=\frac{\tan(n+1)x-\tan x}{\sin x} $$ For $n=1$, the statement ...
0
votes
2answers
51 views

Prove for integers $a,b\ge 3$ that $a^2+b^2+1>(a+1)(b+1)$

Prove for integers $a,b\ge 3$ that $a^2+b^2+1>(a+1)(b+1)$. Original question asked for positive real solutions, but I've changed it to integers. It's question I've come up with. AM-GM ...
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3answers
39 views

Can mathematical inductions work for other sets?

I know that induction works only for the natural numbers $\mathbb{N}$. We first have to prove the base case. And we then prove that if the statement $p(k)$ holds then $\color{blue}{\textbf{p(k+1)}}$ ...
2
votes
2answers
47 views

Proof By Contradiction [?]

There are $n$ islands with $n$ bridges connecting pairs of islands (where $n\ge 2$). Prove that some sequence of distinct bridges forms a loop. __ Since it isn't obvious how to prove it directly I ...
2
votes
3answers
186 views

How to prove a sum of series

How do I prove that for any natural number $n$ we have $$\sum_{i=0}^n i^4 \neq \left(\sum_{i=0}^n i\right)^3?$$ Any help would be greatly appreciated.
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1answer
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Help with a demonstration with triangle numbers

Recently I've been doing some work with trianglular numbers. Basically I wanted to show that for every nth triangular number $T_n$ $$T_n=\frac{n(n+1)}2$$ For me the simplicity if this equation is ...
2
votes
3answers
74 views

Inductive Proof that $k!<k^k$, for $k\geq 2$.

Call $P(k): k!<k^k$, for $k\geq 2$ Test it out with 2, and it's true ($2<4$). Assume that $P(k)$ is true for some $k\geq2$. Then show that $P(k+1)$ is true. $P(k+1): (k+1)!<(k+1)^{k+1}$ ...
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2answers
24 views

Help with Factorial Inequality Induction proof

So I'm asked to prove $3^n + n! \le (n+3)!$, $\forall \ n \in \mathbb N$ by induction. However I'm getting stuck in the induction step. What I have is: (n=1) $3^{(1)} + (1)! = 4$ and $((1) + 3)! = ...
2
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3answers
109 views

Proof verification for proving $\forall n \ge 2, 1 + \frac1{2^2} + \frac1{3^2} + \cdots + \frac1{n^2} < 2 − \frac1n$ by induction

Prove by mathematical induction: $\forall n \ge 2, 1 + \frac1{2^2} + \frac1{3^2} + \cdots + \frac1{n^2} < 2 − \frac1n$ Basis Step: (We want to show, $P(2)$, which is 1 + ...
1
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3answers
44 views

Simple Induction Proof

How would one go about proving that $$0<\frac{n}{n+1}<1$$ by mathematical induction? If $p(n)$ is the statement as above, then I know we show $p(1)$, and assume $p(n)$, but in this particular ...
2
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0answers
42 views

Induction Proof on String

Formally prove the correctness of the union construction as follows. Let $M_1$ and $M_2$ be the two $\lambda$-NFA's constructed for $R_1$ and $R_2$ and let $N$ be the $\lambda$-NFA constructed so ...
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0answers
25 views

Proving the principle of definition by generalized recursion using the inductive closure of an induction system

I'm working through Hinman's Fundamentals of Mathematical Logic in order to review some things, and got stuck in an exercise from section 1.2. Specifically, he asks us to prove (what he calls) the ...
5
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1answer
67 views

Proving $\frac {2n}{(a+b)^n} \le \frac {1}{a^n} + \frac {1}{b^n}$ for $a,b>0, n\in\mathbb{N}$ by induction

prove using induction: $$ \frac {2n}{(a+b)^n} \le \frac {1}{a^n} + \frac {1}{b^n} $$ $$a,b \gt 0 , n \in N$$ my attempt: base $n=1$: $$ \frac {2}{(a+b)} \le \frac {1}{a} + \frac {1}{b}$$ ...
2
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3answers
38 views

Proving that $8^n-2^n$ is a multiple of $6$ for all $n\geq 0$ by induction

I have the following induction problem: $8^n-2^n$ is a multiple of $6$ for all integers $n\geq 0$. So far this is what I've done: Base case: $n = 0$ $8^0-2^0 = 6$ $1 - 1 = 6$ $0 = 6$ This ...
0
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0answers
22 views

Relationship between inductive reasoning and first order reasoning [on hold]

I know what is induction and tableau reasoning. I happen to see that if reasoning is done via induction, then the reasoning is not first order. Why inductive reasoning and first order reasoning are ...
2
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4answers
112 views

Prove that $371\cdots 1$ is not prime.

Prove that $371\cdots 1$ is not prime. I tried mathematical induction in order to prove this, but I am stuck. My partial answer: To be proved is that $37\underbrace{111\cdots 1}_{n\text{ ...
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1answer
17 views

show that $\neg(p_1 \vee p_2 \vee… \vee p_n)$ is equivalent to $\neg p_1 \wedge \neg p_2 \wedge… \wedge \neg p_n$ by induction

Use mathematical induction to show that $\neg(p_1 \vee p_2 \vee... \vee p_n)$ is equivalent to $\neg p_1 \wedge \neg p_2 \wedge... \wedge \neg p_n$ whenever $p_1,p_2,...,p_n$ are propositions. So ...
0
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1answer
26 views

Theorem? For every $f:\mathbb{R}\to\mathbb{R}$, for every $A \subseteq R$ where $A$ is finite, $\exists c\in\mathbb{R}:\forall x\in A:(f(x) = c)$.

Your mathematical sense problably twitched when you read the title, as a simple counterexample of the theorem is some one-to-one function. Where then, is the mistake in this proof? Let ...
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2answers
61 views

Prove by induction $n! > n^2$

I am trying to prove the inequality in the title for $n\geq 4$; however, I am stuck on the induction step! Any help would be appreciated. For $n\ge 4$, prove that $n! > n^2$. Base Case: $n=4$, ...
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3answers
54 views

Prove that $1 \cdot 1!+2 \cdot 2!+\cdots+n \cdot n!=(n+1)!-1$

Prove that $1 \cdot 1!+2 \cdot 2!+\cdots+n \cdot n!=(n+1)!-1$ whenever $n$ is a positive integer. Basis step: $P(1)$ is true because $1 \cdot 1!=(1+1)!-1$ evaluate to $1$ on both sides. Inductive ...
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1answer
34 views

Mathematical Induction problem involving binomial coefficients [closed]

I cannot solve it by using mathematical induction. Please help me with it. $$\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+\ldots+\binom{n}{n}=2^n$$ ...
1
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1answer
46 views

Prove by Induction - Sequence

The sequence $x_1, x_2, x_3, \ldots$ is such that $x_1 = 1 $ and $$x_{n+1} \space = \frac{1+4x_n}{5 + 2x_n}$$ Prove by induction that $x_n > 0.5$ for all $n \ge 1$. I have absolutely no clue how ...
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votes
5answers
147 views

Proof writing: how to write a clear induction proof?

What is an effective way to write induction proofs? Essentially, are there any good examples or templates of induction proofs that may be helpful (for beginners, non-English-native students, etc.)? ...
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2answers
29 views

Use induction to figure out the number of handshakes in a party

Every arriving guest shakes hand with everybody else at a party. If there are n guests in the party, how many handshakes were there? Proof by using induction. My approach to this problem was to write ...
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1answer
54 views

Prove summations are equal

Prove that: $$\sum_{r=1}^{p^n} \frac{p^n}{gcd(p^n,r)} = \sum_{k=0}^{2n} (-1)^k p^{2n-k} = p^{2n} - p^ {2n-1} + p^{2n-2} - ... + p^{2n-2n}$$ I'm not exactly sure how to do this unless I can say: ...
0
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1answer
32 views

proof some inequality by induction

I got to proof the following in-equality by induction for an assignment but having a hard time. $$ \frac{2n}{(a+b)^n} \leq \frac{1}{a^n} + \frac{1}{b^n} $$ $a,b > 0$ Thanks in adavance!
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1answer
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“Cascade induction”?

I refer to this answer. The answer is based on several simplification steps, all of them proven by induction. $S_n = 2903^n - 803^n - 464^n + 261^n$ $T_n = 2642\cdot2903^n - 542\cdot803^n - ...
1
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1answer
31 views

Closed Form Summation Example

$$ \sum_{i=1}^n (ai +b) $$ Let $n \geq 1$ be an integer, and let $a,b > 0$ be positive real numbers. Find a closed form for the following expression. In other words you are to eliminate the ...
2
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1answer
43 views

Proof By Induction - $n^2 = \sum_{i=1} ^{n} (2i-1)$ for all $n\geq 1$ [duplicate]

Using Proof By Induction I am trying to prove the following: $n^2 = \sum_{i=1} ^{n} (2i-1) $ for all $n\geq 1$ Here is my solutions so Far: Base Case: $n=1, LHS: 2(1)-1 = 1, RHS = 1^2 = 1, True$ ...
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4answers
183 views

Proving that $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}>\frac{13}{24}$ by induction. Where am I going wrong?

I have to prove that $$\frac{1}{n}+\frac{1}{n+1}+\dots+\frac{1}{2n}>\frac{13}{24}$$ for every positive integer $n$. After I check the special cases $n=1,2$, I have to prove that the given ...
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0answers
20 views

Show that $f(b^i n) \le c^i f(n)$

Let $f$ be a b-smooth function. Let $c$ and $n_0$ be constants such that $f(b n) \le c f(n)$ $\forall $ $n \ge n_0$. Show that $\forall $ $ i \in \mathbb{N}, f(b^i n) \le c^i f(n)$ I thought I should ...
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0answers
32 views

Prove, by induction that $a_{n} > \frac{1}{2^n}$ for all $n \in \mathbb{N}$ [closed]

$$a_{1} = \frac{5}{2} $$ $$a_{n+1} = \frac{1}{2}(a_{n} +2)~\forall~n\in\mathbb{N} $$ Prove, by induction that, $$a_{n}\gt \frac{1}{2^n}~\forall~n\in\mathbb{N}$$
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3answers
45 views

$\sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n}$ [duplicate]

Prove that for $n\geq 2, \: \sum_{k = 1}^{n} \frac{1}{k^{2}} < 2 - \frac{1}{n} $ I used induction and I compared the LHS and the RHS but i'm getting an incorrect inequality
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1answer
27 views

Induction Mathematics and Factorials

\usepackage{amsmath} Evaluate the sum $\sum_{k=1}^{n} {k\over (k+1)!}$ $\sum_{k=1}^{1} {1\over (1+1)!} = {1\over 2}$ $\sum_{k=1}^{2} {2\over (2+1)!} = {5\over 6}$ $\sum_{k=1}^{3} {3\over (3+1)!} ...
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2answers
33 views

Proofing Induction Mathematics

I have just started to cover induction mathematics in my Discrete Mathematics class and I'm a little confused as to where to go with this problem. Am I on the right track? Prove that 9 divides (n^3 ...
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2answers
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Proving that $F_{kn}$ is a multiple of $F_n$ by induction on $n$ (Fibonacci numbers)

Question: I want to prove that $F_{kn}$ is a multiple of $F_n$. Approach: I have to deduce this result from the following results: $$F_{n+k} = F_{k}F_{n+1} + F_{k-1}F_{n}$$ I have shown the ...
3
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2answers
39 views

Proving $1+5+9+\cdots+(4n+1) = (n+1)(2n+1)$ by induction (is there a typo?)

Using mathematical induction, prove that $$1+5+9+\cdots+(4n+1) = (n+1)(2n+1).$$ I understand the steps to take in order to prove by induction. It is also to my understanding that step 1 would be ...
4
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2answers
62 views

Proving that $3^n<n!$ when $n\geq 7$

It's been 10 years since my last math class so I'm very rusty. How would I go about proving $$3^n < n!$$ where $n \geq 7$? I understand that factorials grow faster than set values with a variable ...
2
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4answers
77 views

Rewrite $\sum_{i=0}^{n-1} (2i+1)=n^2$ to start induction from $k = 0$?

I'm trying to learn mathematical induction. The text asks for being totally rigorous i.e start induction from $k=0$. I want to prove that $$\sum_{i=0}^{n-1} (2i+1)=n^2,$$ i.e. the sum of the first ...
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2answers
51 views

Construction of Natural Numbers

I am trying to prove that the natural numbers can be constructed from the product of a power of $2$ and an odd number. For all $n \neq 0$ in the natural numbers, $n = (2k+1)(2^p)$, where $k$ and $p$ ...
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Prime numbers proof [closed]

Statement: Let n be a positive integer. There exists a prime number greater than n. Proof: Consider m = n! + 1. We know that m is divisible by some prime p. But no number between 2 and n is a ...
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4answers
202 views

Proving $\sqrt{1+\sqrt{2+\cdots+\sqrt{n}}} < 3$ for $n\geq 1$ by induction

How can I prove by induction that $\sqrt{1+\sqrt{2+\cdots+\sqrt{n}}} < 3$ for $n\geq 1$? My guess is that there must be another form to express the sum of nested square roots, but I don't know how ...
4
votes
1answer
41 views

Prove that if $p \mid a_1a_2 \ldots a_n$, then $p \mid a_j$ for some $j$ with $1 \le j \le n$

Let $p$ be a prime number and $a_1, a_2, \ldots, a_n$ be integers. Prove that if $p \mid a_1a_2 \ldots a_n$, then $p \mid a_j$ for some $j$ with $1 \leq j \leq n$. The hint was to use induction. ...
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3answers
44 views

Find a formula for $ 1\times3^0 + 3\times3^1 + 5\times3^2 + … +(2n+1)\times3^n$.

The original exercise is to find a formula for this and prove it via induction. However, I am having a problem deriving such a formula. How do you normally approach this types of problems, is it a ...
0
votes
1answer
20 views

How to go about induction that deals with inequalitites

The only thing i've been able to do is to prove it for 1. How do i go about prvoing it for k+1?
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1answer
28 views

Nim Variant - Strong Induction Proof

Here we will play a variant of Nim where there is an additional move option in some cases. If two or more piles have the same number of stones, a player may remove the same number of stones from ...
0
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1answer
29 views

How to perform induction step in this question?

Q:Prove By induction $2^{n+1} > n^2$ for all positive integers. Step 1: Base case: $n=1$, we get $4>2$ Step 2: Induction hypothesis: $n=k, 2^{k+1} > k^2$ Step 3: Induction Step: to ...
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1answer
110 views

Proving by induction on the number of vertices that: every acyclic simple graph is bipartite

Prove that every acyclic simple graph is bipartite, by the use of induction. I have quite some trouble with induction. Specifically, I know that acyclic graphs have at least one vertex that has a ...