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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How to prove $n < \left(1+\frac{1}{\sqrt{n}}\right)^n$

I want to know how to prove the following inequality. For $n = 1, 2, 3, \ldots $ $$ n < \left(1+\frac{1}{\sqrt{n}} \right)^n $$ I tried with math induction but I failed.
2
votes
1answer
570 views

Combinatorics - Catalan numbers recursive function proof by induction help

I'm trying to prove that $$C_n = \sum_{i=0}^{n-1} C_iC_{n-i-1}$$ when $1\leq n$ and $C_0 = 1$ by induction. ($C_n$ is the $n$th catalan number). For $n=1$ this is true. Assume it is true for $n=k$: ...
0
votes
1answer
30 views

Proving Cauchy-Schwarz related proof using induction

So the first thing I was asked to prove was this: If $a_1,a_2,...,a_n$ and $b_a,b_2,...,b_n$ are real numbers, use induction to show. ...
0
votes
3answers
36 views

Proof by minimum counter example

I need to prove that $n^4-n^2$ is divisible by 12 by minimum counter example. I understand the process but I don't understand how we arrive at m>=7. I have seen different proofs but I still don't know ...
3
votes
2answers
25 views

Proof by induction for $ \sum_{n}^{M} \cos(2n) = \frac{\sin(M) \cos(M+1)}{\sin(1)} $

Can someone show me an induction for $$ \sum_{n}^{M} \cos(2n) = \frac{\sin(M) \cos(M+1)}{\sin(1)} $$? My problem is doing that induction with $M$, I am not sure how to proceed to get the right side of ...
3
votes
2answers
51 views

Show that it is the solution of the recurrence

I have to show that the solution of the recurrence $$X(1)=1, X(n)=\sum_{i=1}^{n-1}X(i)X(n-i), \text{ for } n>1$$ is $$X(n+1)=\frac{1}{n+1} \binom{2n}{n}$$ I used induction to show that. I have ...
3
votes
3answers
78 views

prove by induction that $29^n - 21^n$ is always divisible by $8$

I have to prove by induction that that $\forall n \in N,$ $8 | (29^n - 21^n) $ . I understand how to prove things with induction generally, but im not sure where to even start with this one. I ...
0
votes
2answers
39 views

Proof using Induction

Give the induction proof of: $$ 1.2 + 2.3 + k(k+1) = \frac{k(k+1)(k+2)}{3} $$ Is this proof even possible? Not sure how to do.
2
votes
0answers
55 views

Prove, using the method of mathematical induction that the following holds true

For natural numbers $n\ge1$ show the following inequality using induction. $$n^{1/n}\le 1+\sqrt{\frac{2}{n}}$$
0
votes
1answer
22 views

Proving number of edges in F = n - k

So if we let F = (V,E) be a forest with n vertices and k connected components (trees), how can I prove that the number of edges in F = n - k ? I was thinking of using induction, but I'm super lost. ...
3
votes
4answers
59 views

Prove that $F(1) + F(3) + F(5) + … + F(2n-1) = F(2n)$

(These are Fibonacci numbers; $f(1) = 0$, $f(3) = 1$, $f(5) = 5$, etc.) I'm having trouble proving this with induction, I know how to prove the base case and present the induction hypothesis but I'm ...
1
vote
1answer
14 views

Looking for a way to improve my inductive proof of a statement derived by Rolle's Theorem

The following problem is 'absolutely' clear: Problem: Let $f$ be continuous on the interval $[a,b]$ and $n$-times differentiable on $(a,b)$ and $f$ vanishes on $n+1$ points $x_0< x_1 < \dots ...
1
vote
0answers
20 views

Prove that if there are $2n$ points and $n^2+1$ straight lines connecting them, then there are at least $n$ triangles in this shape.

Proof by induction. For $n=2$, it says that if we have $2(2)=4$ points and $2^2+1=5$ lines connecting them to each other, then there are at least 2 triangles in this shape. Which is true (shown ...
1
vote
1answer
28 views

Inequalities - proof by induction

Proof by induction involving inequalities completely escapes me. I've encountered the following problem: For which non-negative integers n is $n^2 ≤ n!$? Prove your answer (by induction). So, ...
12
votes
5answers
717 views

Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction

How can I prove that $$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$ for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction. Thanks
1
vote
3answers
58 views

Using mathematical induction to show that for any $n\ge$ 2 then $\prod_{i=2}^n\left(1-\frac{1}{i^2}\right)=\binom{n+1}{2 \cdot n}$

I'm trying to work through some practice problems but I've been stuck on this for god knows how long now and I've no idea where to even start. Just wondering if it would be possible for someone to ...
1
vote
1answer
59 views

Induction Proof with Combinations?

Show that for all $n\geq0$ $$\binom{n}{0}3^n+\binom{n}{1}3^{n-1}+\dotsc+ \binom{n}{n-1}3^{1}+\binom{n}{n} $$ $$= \binom{n}{0}5^n-\binom{n}{1}5^{n-1}+\binom{n}{2}5^{n-2}-\binom{n}{3}5^{n-3}+\dotsc ...
1
vote
3answers
49 views

Induction Proofs - Mathematics

How do I show by mathematical induction that $2$ divides $n^2 - n$ for all $n$ belonging to the set of Natural Numbers
0
votes
3answers
67 views

Prove that $(1+2+3+\cdots + n)^2 = 1^3+2^3+3^3+\cdots + n^3$ for every $n \in \mathbb{N}$ [duplicate]

Prove that $(1+2+3+\cdots + n)^2 = 1^3+2^3+3^3+\cdots + n^3$ for every $n \in \mathbb{N}$. Proof. We will use mathematical induction. If $n = 1$, then we have $(1)^2= 1^3 = 1$. We must show that ...
0
votes
0answers
22 views

different representations of strong induction

I've seen 2 forms of strong induction; just wondering how one follows from the other. $1) f(n_0)\wedge f(n_1)\wedge\cdots \wedge f(n_{k-1})\wedge f(n_k)\wedge \forall_n[f(n-k)\wedge ...
2
votes
2answers
55 views

How to prove $0 < a_n < 1$ by induction

I know $n \in \mathbb{N}$ and... $$ a_n = \begin{cases} 0 & \text{ if } n = 0 \\ a_{n-1}^{2} + \frac{1}{4} & \text{ if } n > 0 \end{cases} $$ Base Case: $$a_1 = ...
21
votes
7answers
519 views

Illustrative examples of a phenomenon in the logic of mathematical induction

It is said (and I myself have said) that in some cases the easiest way to prove a statement by mathematical induction is to prove a stronger statement by mathematical induction, because then one has a ...
1
vote
1answer
25 views

Showing that a sequence (defined in terms of the previous sequence term) is increasing and bounded above

I'm stuck on this problem and I was wondering if you would be kind enough to help. The question follows: Let $x_{1} = 1$ and $x_{n}$ = $\sqrt{ 1 + 2x_{n-1}}$ for n $\geq$ 2. Show that the sequence ...
1
vote
3answers
27 views

Show that $7\mid 2^{3n} -1 \,\,\,\forall n\in \mathbb N^+$

Show that $7\mid 2^{3n} -1 \,\,\,\forall n\in \mathbb N^+$ Should I prove this by induction? If so, how should I go about it?
5
votes
4answers
59 views

Prove that $ \frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!}+\cdots + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!}$ for $n\in \mathbb N$

I want to prove that if $n \in \mathbb N$ then $$\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!}+ \cdots+ \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!}.$$ I think I am stuck on two fronts. First, I don't know ...
2
votes
3answers
88 views

How to prove this statement: $\binom{r}{r}+\binom{r+1}{r}+\cdots+\binom{n}{r}=\binom{n+1}{r+1}$

Let $n$ and $r$ be positive integers with $n \ge r$. Prove that Still a beginner here. Need to learn formatting. I am guessing by induction? Not sure what or how to go forward with this. Need help ...
1
vote
1answer
40 views

Induction proof for Fibonacci numbers

I am trying to get a hang on the induction method for proof, but I'm still dubious of many aspect of this proof regarding its application to sequences of integers, such as the Fibonacci sequence. ...
0
votes
2answers
28 views

Prove sum of combinations

Let n and r be positive integers with n ≥ r. Prove that C(r, r) + C(r + 1, r) + ... + C(n, r) = C(n + 1, r + 1) I would like to approach with mathematical induction. However, I don't understand what ...
1
vote
3answers
27 views

Proving this $F_{n+1} \cdot F_{n-1} - F^2_n = (-1)^n$ by induction

Where $n \in \mathbb{N}$ and $$ F_n = \begin{cases} 0 & \text{ if } n = 0 \\ 1 & \text{ if } n = 1 \\ F_{n-1} + F_{n-2} & \text{ if } n > 1 ...
0
votes
4answers
42 views

Prove $H_{2^m} \leq 1 + m$, where $H_n = \sum\limits_{k=1}^n \frac{1}{k}$

I really I am not seeing how to continue my approach, which is this. Base case: $m = 1$, so we have $H_2 \leq 2$, where $H_2 = \sum\limits_{k=1}^2 \frac{1}{k} = \frac{1}{1} + \frac{1}{2} = ...
2
votes
4answers
56 views

Equilateral triangle is cut in $4^n$ congruent equilateral smaller triangles

I have an assignment on proof by induction: Suppose n is a positive integer. An equilateral triangle is cut into $4^n$ congruent equilateral triangles, and one corner is removed. (Figure 1 ...
0
votes
1answer
21 views

Strong induction on a sequence, proving two functions are equal?

Excuse the poor title, but my understanding is still a little fuzzy. Admins feel free to change it Here is the question from the book. suppose that $f_{0}, f_{1}, f_{2}...$ is a sequence defined ...
1
vote
2answers
69 views

Proving a combinatorics equality

How to prove the following? Should I use induction or something else? Let n and r be positive integers with n ≥ r. Prove that $${\binom{r}{r}} + {\binom{r+1}{r}} + · · · + {\binom{n}{r}} = ...
2
votes
1answer
24 views

Induction proof with inequalities

Consider the following claim: $$5^n > 4^n + 3^n + 2^n$$ (a) For what natural numbers is this claim true? (b) Prove that your answer to (a) is correct using induction on n.
0
votes
0answers
22 views

A couple of questions about induction.

I am trying to understand the proof of the theorem 56.4 on page 387. I have asked a related question before and according to user Matt S, we use strong induction here. Where in the body of proving ...
0
votes
3answers
37 views

By induction, show that for ∀n∈N, it is true that: [duplicate]

$$\sum_{i=1}^n 2^i=2+2^2+2^3+.....+ 2^n=2(2^{n}-1)$$ Any help/explanations would be REALLY appreciated. Also in the same vein: By induction, show that $$∀n∈\mathbb N: 11^{n+2} + 12^{2n+1}$$ is ...
0
votes
1answer
16 views

Lining a rectangular building square panels

I've been working on this for what feels like a lifetime now and I'm just not getting anywhere with it. I'm wondering if someone would be able to explain how to solve it for me? There is a ...
2
votes
1answer
62 views

Proving terms are rational using Mathematical Induction

I was able to do the first part of the question, in the second part (Proof by Induction), I showed it holds for $n=1$: Then I Assumed its true for $n=k$. ...
2
votes
0answers
25 views

Confusion on particular step in this weak induction proof

I am studying for a midterm, and I came across this proof. Use mathematical induction(weak) to show that for all integers n $\geq$ 2 , if x$_{1}$, x$_{2}$, ... x$_{n}$ are strictly between 0 and 1 ...
21
votes
8answers
380 views

Examples where it is easier to prove more than less

Especially (but not only) in the case of induction proofs, it happens that a stronger claim $B$ is easier to prove than the intended claim $A$ (e.g. since the induction hypothesis gives you more ...
0
votes
1answer
21 views

An induction proof in a set.

I have an induction problem that I have no idea how to start. So the question goes like this. Let $x_1=1$, $x_2=2$ and $x_n=x_{n-1} + 2x_{n-2}$. Prove that $x_n=2^{n-1}$ for all $n$ in the natural ...
-1
votes
1answer
50 views

Prove by mathematical induction [closed]

Prove by mathematical induction this expression : $$\frac 1{\sqrt{n+1}+\sqrt {n}} \lt \frac 1{2\sqrt{n}}$$
1
vote
1answer
74 views

What sets does $\mathbb{N}$ include?

My text states that the set $\{1, 2, 3...\}$, and the set $\{101, 102, 103, 104...\}$ are elements of $\mathbb{N}$. Doesn't this imply that $\mathbb{N}=\{1, 2, 3... 101, 102, 103, 104...\{1, 2, 3 ...
2
votes
1answer
54 views

Structural Induction, Propostitonal formulae problem

I am kind of overwhelmed by this question. Can anyone give me some hints about where to start? Propositional formulae PF are inductively defined over the Boolean constants B := {1, 0} (true and ...
0
votes
1answer
26 views

induction with factorials

I need help with this please. I understand step one is to let $n=1$. step two let $ n = k$. Step three prove for $k+1$. But I would like a clear example of each... Prove $$\sum_{i=1}^n ...
1
vote
0answers
12 views

Induction for quantified statement with two discrete parameters

Given a quantified statement ∀n, n>0 (∃x, x>2k | x=2k+n) ( a subset of the natural numbers) This can logically this can be deduced as valid; however, I wish to use induction. Specifically I would ...
1
vote
2answers
27 views

Induction of factorial

I was perusing the wikipedia page on Mathematical induction, and it mentions it's possible to prove by induction that. $\frac{n^{n}}{3^{n}}<n!<\frac{n^{n}}{2^{n}}$ for $n\geq6$ Proof for $n=6$ ...
0
votes
4answers
31 views

Use strong induction to prove that n$\leq$3$^{n/3}$ for every integer n$\geq$0

Use strong induction to prove that n$\leq$3$^{n/3}$ for every integer n$\geq$0. According to steps of Strong Induction, 1) I assume the predicate as P(n): n$\leq$3$^{n/3}$ for every integer ...
-5
votes
0answers
39 views

Induction and limit cases

Say we want to use induction as our proof. We prove the base case $K=1$ and we show the inductive step that if a statement is true for $K$ then it is also true for $K+1$. Is this enough to show that ...
1
vote
1answer
32 views

Proof by induction of whole numbers

A sequence $X_1, X_2,\dots,X_n$ is defined by: $X_1 = 1$ and $X_{k+1} = \dfrac{X_k}{X_k + 2}$ for $k\ge1$. Show by using induction that $X_n = \dfrac1{2^n - 1}$ for all $n\ge1$. So far I've showed ...