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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Mathematical Induction [closed]

Prove the following using the principle of mathematical induction for all n belonging to positive integers.
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Help with induction proof for recurrent function

I am having issues with the following inductive proof. Prove by induction on $n$ that $$ a(n) = n!\bigg(\frac{1}{0!} + \frac{1}{1!} + \cdots + \frac{1}{(n-1)!}\bigg)$$ for all $n \geq 1,$ where ...
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Coming up with an alternative proof by induction

Kindly refer to Q4 of this handout. "$2n$ dots are placed around the outside of the circle. $n$ of them are colored red and the remaining $n$ are colored blue. Going around the circle clockwise, you ...
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53 views

Show by induction whether $1+\frac{1}{2}+\frac{1}{3}\cdots+\frac{1}{n}$ is an integer or not

How can I show by induction whether $1+\frac{1}{2}+\frac{1}{3}\cdots+\frac{1}{n}$ is an integer or not? Progress : For n=1 the expression is $(=1)$ an integer. How can I show the next step?
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Prove by Induction $64\mid (7^{2n} + 16n − 1)$

We have to show by Mathematical Induction that $64\mid (7^{2n} + 16n − 1).$ Progress : Let us suppose $P(n)$ be the statement i.e., $P(n): 64\mid(7^{2n} + 16n − 1)$ For $n=1$, $(7^{2\cdot ...
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1answer
31 views

Proof by induction (power rule of the derivative)

Using the differentiation formulas $\displaystyle\frac{d}{dx}x=1$ and $\displaystyle\frac{d}{dx}(fg)=f\frac{dg}{dx}+g \frac{df}{dx}$, prove that $$\frac{d}{dx} x^n=nx^{n-1}$$ for all natural number ...
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Proof by Induction Inequalities

Use the PMI to prove the following for all natural numbers: $3^n ≥ 1+2^n$. I have already verified the base case but am having trouble doing so with the inductive case. Thanks!
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RE : Is greatest common divisor of two numbers really their smallest linear combination?

This is in reference to the same proof given in the post Is greatest common divisor of two numbers really their smallest linear combination? I couldn't add a comment there so I'm asking it here. I ...
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30 views

Prove that this is true

$$\sum\limits_{i=1}^{n}i^x = P_{x+1}(n)$$ Let x be any nonnegative integer and show that there is a polynomial $P_{x+1}$ of degree $x+1$ for every $n$ greater than or equal to $1$. ...
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1answer
31 views

How do I use complete induction here?

Suppose currency consists of 3 and 4 cent coins. Suppose you want to buy an item that is worth 9 cents. Show that if you have an unlimited number of 3 and 4 cent coins you can buy anything greater ...
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Mathematical induction used on Fibonacci Sequence

I have no clue how to go about doing this question using induction. It states that the Fibonacci sequence is defined as: F0 = 0 F1 = 1 Fn = Fn-2 + Fn-1 for n>=2 Let S(n) = Fo + F1 + F2 +...+ ...
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4answers
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How can I prove this inequality by M.I. or otherwise?

This question is from the past examination: $$f(n)=\frac{2}{3}(n^{3/2}-1)+\sqrt{n}$$ $$g(n)=1+\frac{2}{3}((n+1)^{3/2}-2^{3/2})$$ My task to prove $f(n)≥g(n)$ for all $n≥1$. I have tried M.I here. ...
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2answers
29 views

Factorials and Mathematical induction

I'm having a bit of trouble understanding mathematical induction, particularly when there's a question with powers or factorials. For example I have a problem 1 x 1! +2 x 2! + 3 x 3! +... + n x n! = ...
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1answer
29 views

Proof of AM-GM Inequality (setting $a_n$ in the last step) [duplicate]

I have been reading this and this, but I don't understand how one of the step works. Let $a_n=\frac{a_1+a_2+\cdots+a_{n-1}}{n-1}.$ How do you set $a_n$ to meet certain criteria and not lose ...
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2answers
26 views

Is there some trick to manipulating an equation? (adding 0s, multiplying by 1, etc..)

I have such a hard time doing this sort of thing that it's annoying me. I'm not very mathematically inclined but it frustrates me that a solution with such a small answer takes me more than a page to ...
0
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2answers
38 views

An induction question on showing that eventually $(n+2)^n < (n+1)^{n+1}$

Show that eventually $(n+2)^n < (n+1)^{n+1}$ I can see that this is obvious by evaluation at n>2, but I am having a hard time separating to get the induction step within the parenthesis. I am ...
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3answers
109 views

Counting the number of different ways in which groups of one or two can be formed…

I'm having trouble proving that the number of ways n>3 people can be divided into groups of either one or two is equal to: $A_n = A_{n-1} + (n-1)⋅A_{n-2} $ I'm trying to prove this by counting but ...
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0answers
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Do I have to prove it by induction with respect to $n$ or to $k$?

I want to prove by induction, that the solution of the recurrence relation $T(n)=2T \left ( \frac{n}{2} \right )+n^2, n>1 \text{ and } T(1)=1$ is $n(2n-1)$. We have to suppose that $n=2^k, k \geq ...
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1answer
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Finding $a_n$ if $a_0=a_1=1,a_{n+1}=n(a_n+a_{n-1})\ \ (n\ge 1).$

The problem states: Suppose $a_0,a_1,a_2,...$ is a sequence such that $$a_0=a_1=1,\ \ \ a_{n+1}=n(a_n+a_{n-1})\ \ \ (n\ge 1).$$ Guess a formula for $a_n$, valid for $n\ge 1$, and use mathematical ...
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1answer
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Prove $\forall n \in N$, every set of natural numbers of size n has a maximum element. May assume that sets do not repeat numbers.

Prove using induction. So i'm a bit confused about how to do this question. My attempt at it seems like i'm missing a lot and it looked to easy. ...
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2answers
58 views

Can this be proved without making use of derivatives?

Problem: Let $\left(a_{n}\right)$ be a sequence with $a_{1}=1$ and $a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{2}{a_{n}}\right)$. It must be proved that $a_{n}\geq\sqrt{2}$ for $n\geq2$. I have a proof, ...
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1answer
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Prove through structural induction that a binary tree has an odd number of nodes

A full binary tree is a binary tree where every node has either 0 or 2 children. Prove that every non-empty full binary tree has an odd number of nodes. I dont know how to get started with this ...
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3answers
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Induction proof for n > 0

Prove using induction the following: for n > 0, 1 ∙ 1! + 2 ∙ 2! + ..... + n ∙ n! = (n + 1)! - 1 I'm not very good at proving proofs with the induction method, help would be greatly appreciated
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Prove using induction principles

$$\forall{n,a>1}:\;\sum\limits_{k=1}^{2^n-1}\frac{1}{k^a}\;\leq\left(\frac{1-2^{n(1-a)}}{1-2^{1-a}}\right)$$ For any fixed value of $a > 1$. Induction step: $$\sum_{k=1}^{2^{n+1} - 1} ...
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Prove this sum of binomial terms using induction.

Here's the problem stumping me today: Let $n \in \mathbb{N}$ and $r \in \mathbb{N}$ such that $r \leq n$, and prove using induction that $\binom{n+1}{r+1} = \sum\limits_{i=r}^n \binom{i}{r}$. I've ...
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2answers
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Proof by induction steps

Today in class, the instructor is trying to show that for $n \ge 0$, $n < 2^n$. And this are the steps he took: First we assume the inductive hypothesis i.e. $0 < 2^0$, and this is true. Then ...
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1answer
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Do we have to claim it? If so, at which point?

I have to solve the recurrence relation $$T(n)=\left\{\begin{matrix} 3T\left (\frac{n}{4} \right)+n & , n>1\\ 1 &, n=1 \end{matrix}\right.$$ and prove by induction that the solution I ...
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Chaining Exponent Rules Together

I'm having trouble understanding why the following property is true and want to make sense of it before going ahead and use it in my proof by induction: $$2^{2^n}=2^{2^{n-1}}\times 2^{2^{n-1}}=\left( ...
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Prove this induction problem [closed]

Show that every positive integer $N$ less than or equal to $n$ factorial, is the sum of at most $n$ distinct positive integers, each of which divides $n!$.
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Number or regions formed when $n$ points on a circle are joined

The maximum number $R_{n}$ of regions formed when $n$ points on a circle are joined in pairs is $\frac{1}{24}\left(n^{4}-6n^{3}+23n^{2}-18n+24\right)$. This is a fact that I have read in several ...
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Prove $(\log{n})^2\leq 2^n$ by induction

I've trying to solve this for quite a while now, but not being able to finish the proof. Prove using induction that $(\log{n})^2\leq 2^n$
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Inductive definition on a sequence

I have a question which goes like this: "Show the inductive definition for the sequence {$a_n$} if $a_n = 5 + 7n$ and $n = 0, 1, 2, 3, 4, ...$ I was wondering given the formula to find $a_n$ is it ...
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1answer
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Mathematical Induction: Sum of first n odd perfect cubes

The series is $$P_k: 1^3 + 3^3 + 5^3 + ... + (2k-1)^3 = k^2(2k^2-1)$$ and I have to replace $P_k$ with $P_{k+1}$ to prove the series. I have to show that $$k^2(2k^2-1) + (2k-1)^3 = ...
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4answers
218 views

Mathematical Induction Question, Proof Help [duplicate]

Prove using Mathematical Induction that for all natural numbers ($n>0$): $$ \frac 1 {\sqrt{1}} + \frac 1 {\sqrt{2}} + \cdots + \frac 1 {\sqrt{n}} \ge \sqrt{n}. $$ ...
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3answers
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Mathematical Induction: Sum of first n even perfect squares [duplicate]

So the series is $$P_k: 2^2 + 4^2 + 6^2 + ... + (2k)^2 = \frac{2k(k+1)(2k+1)}3$$ and i have to replace $P_k$ with $P_{k+1}$ to prove the series. I have to show that $$\frac{2k(k+1)(2k+1)}3 + ...
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1answer
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Follow-up question on mathematical induction with arbitrary base case

Note: This question has already been answered here Proving mathematical induction with arbitrary base using (weak) induction. I was trying to 'reconstruct' at least one proof given in this question ...
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Proof by Induction for Splay Tree?

I'm preparing for an exam about Trees. One of the questions that appear in Mark Allen Weiss' "Data Structures and Algorithms Analysis in C++" is: Prove by induction that if all nodes in a splay tree ...
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votes
5answers
315 views

polynomial with positive integer coefficients divisible by 24?

I have to show that $n^4+ 6n^3 + 11n^2+6n$ is divisible by 24 for every natural number, n, so I decided to show that this polynomial is divisible by 8 and 3, but I'm having trouble showing that it is ...
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1answer
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Prove inequality formula by induction

my question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook. Page 35. Exercise 1. Prove the following formula by induction: ...
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1answer
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prove if $b \geq a$, then $a^{b} \geq b^a$

I found that if b = a - 1, then $a^{b} \leq b^{a}$ and if a = b, then $a^{b} = b^{a}$ for obvious reasons. Now, i'm having a hard time figuring out how to prove that if $b \geq a$, then $a^{b} \geq ...
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3answers
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prove $m^{m-1} < (m-1)^m$ for m > 3

I found that if m > 3 then $m^{m-1} < (m-1)^m$ for m > 3 seems to hold true for a lot of cases. Can someone prove this inductively ?
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1answer
107 views

Prove that $(n!)^ 2 \gt n^n$ [duplicate]

Prove the above by by mathematical induction By any other method. I was just asked to prove this so I thought of using mathematical induction. My effort : I started first by verification and ...
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1answer
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help with understanding a proof

again I'm stuck already on the first steps of an inductive proof of $$ (a+b)^{n+1} = \sum_{k=0}^{n+1} {n+1\choose k}a^kb^{n+1-k} $$ that is, I'm trying to understand the solution to this. It starts ...
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votes
3answers
37 views

Given the sequence $a_0=1, a_1=2, a_2=3, a_n=a_{n-1}+a_{n-2}+a_{n-3}$, prove by strong induction that for $n\geq 0, a_n \leq 2^n$

I've been trying to work this out for some time and I keep getting stuck. Here is what I have thus far: Base Case: $n=0 ; 1 \leq 1$ $n=1 ; 2 \leq 2$ $n=2 ; 3 \leq 4$ Induction hypothesis: ...
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votes
1answer
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Strong induction on a summation of recursive functions (Catalan numbers)

I've been stuck on how to proceed with this problem. All that's left is to prove this with strong induction: $$\forall n \in \mathbb{N}, S(n) = \sum_{i=0}^{n-1} S(i)*S(n - 1 - i)$$ Some cases: S(0) ...
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1answer
51 views

How to write a general proof to prove that for all $m$, $m^n \geq n^m$

After proving $m^n \geq n^m$ for several values of $m$, it can be inferred that for every $m$ there's a $k$ such that if $n \geq k$, $m^n \geq n^m$. In other words, this can be generalized as: For ...
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How would you prove inequality $2^n \gt n^{10}$ using induction

For the base case I can put a number such as $100$ for $n$ so $2^{100}\gt 100^{10}$. Ok so now the induction hyp: $2^{n+1} > (n+1)^{10}$ for $n \gt 101.$ where do I go from here? Also do I have ...
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MIT OCW Assignment question Strong Induction

There are two types of creature on planet Char, Z-lings and B-lings. Furthermore, every creature belongs to a particular generation. The creatures in each generation reproduce according to certain ...
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can any identity involving integers be proved by mathematical induction

Hello mathematics community, Today I was studying mathematical induction which is an axiom. I was wondering Can "ANY" identity or inequality involving integers which is already proven can also be ...
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2answers
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I need to prove that $Z^{n}=r^{n}(\cos n\theta +i\sin n\theta)$ is true by Induction. Can someone confirm whether I have done it right or not?

The equation $$Z^{n}=r^{n}(\cos(n\theta) +i\sin(n\theta))\space\space\space\space\space\space\space\space (1)$$ has to be proved by induction. It is given that $$Z=r(\cos(\theta) ...