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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174 views

After removing any part the rest can be split evenly. Consequences?

Let $S$ be a finite collection of real numbers not necessarily distinct . If any element of $S$ is removed then the remaining real numbers can be divided into two collections with same size and same ...
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0answers
36 views

Inductive proof of the property $f(k+2)=f(k)+f(k+1)$ for the numbers given in terms of the golden ratio [duplicate]

Help prove through induction that $f(k+2)=f(k)+f(k+1)$ using the golden ratio $\frac1{\sqrt5}\phi^n-\frac1{\sqrt5}\left(\frac{1-\sqrt5}2\right)^n$ $F_{n+2}=F_n+F_{n+1}$, using golden ration $f_n = ...
0
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1answer
84 views

Induction in a first order system with ZF

Suppose I have some characterization of the natural numbers $N$ in a first-order system under ZF. To be precise, I have $N = \lbrace n: \forall w:( w\space is\space inductive) \rightarrow n \in w ...
1
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1answer
79 views

Proving a Recurrence Relation by induction

I have the Recurrence Relation: $ T(n)=T(log(n))+O(\sqrt{n}) $, and I'm being asked to prove by induction an upper bound. I'm also allowed for ease of analysis to assume $n=2^m$ for some $m$. So here ...
1
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1answer
26 views

How to prove that at Complete Binary Tree (CBT) at height $h$ we have $2^h$ leaves

I try to prove it by induction, please tell me if I'm right... The induction assumption - For every CBT at height $h$ there is $2^h$ leaves. The base of the induction is right (I'm writing this proof ...
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3answers
133 views

Set of palindromes with induction

Let $A = \{a_1, a_2, ..., a_k\}$ be a finite alphabet. a. Define, using structural induction, set of all palindromes of A. b. Find the recurrent pattern which represents the number of all ...
0
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1answer
56 views

Proof of the number of the leaves in a full binary tree

I need to proof by induction that at full binary tree there are $\frac{n+1}{2}$ leafs if $|V|=n$. So, I won't write you the whole proof, just my idea, and I'd like to know if this OK... So we ...
1
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4answers
82 views

Help proving $9^n-8n-1$ is divisible by $8$ for all $n > 1$ by induction

I have been trying to prove that $9^n-8n-1$ is divisible by $8$ for all $n$ integers greater than 1. My progress: Let $n = 2$. This gives us the expression equal to $64$ which is a factor of 8. Now ...
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2answers
75 views

Proof by induction that certain number is an integer

Prove that the number $\frac{2n^5}{5} + \frac{n^4}{2} - \frac{2n^3}{3} - \frac{7n}{30}$ is an integer $\forall n \in \mathbb{N}.$
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1answer
23 views

Questions about k elements subset of an n elements set.

I need to prove by induction that the number of 2-elements subset of an n elements set is $\frac{n(n-1)}{2}$ I am stuck on where I should start from and how should I solve this. I am guessing that ...
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3answers
203 views

Puzzle: “Yes colour of my hat is white.”

There are $n$ people in room each being put on hat from amongest at least $n$ white hats and $n-1$ black hats. They stand in a queue, so that everyone can see the colour of the hat of the person ...
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1answer
48 views

Proof of sum of binomials over upper index (induction)

How would you proof $$ \sum_{m=k}^{n}\binom{m}{k} = \binom{n + 1}{k + 1} $$ with $n \geq k$ and $n$, $k \in \mathbb{N}$ by induction? I had some approaches but wasn't sure if they were right, so I'd ...
2
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2answers
43 views

Mathematical Induction Angles proof.

![this is a very dicy problem. It would be great to go into details of how to prove it using induction or any other alternate way is highly appreciated.][1] ...
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1answer
28 views

Finding $\sup$ and $\inf$ of $\{\frac{nk}{1+2n+3k} : n,k \in \Bbb{N}\}$

I'm trying to solve the following problem: Find $\sup$ and $\inf$ of $A=\{\frac{nk}{1+2n+3k} : n,k \in \Bbb{N}\}$ and maximal and minimal element of this set. As for $\sup(A)$ and $\max(A)$ I tried ...
0
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3answers
46 views

A Problem involving simple mathematical induction.

$6^n-5n+4$ is divisible by $5 \;$ for all natural numbers $n$. what I did is: IA $A(1):\;6^1-5\cdot1+4=5$ which is true. IS $A(n):\; 6^n+5n+4$ is also divisible by $5$. Show $A(n+1)$ is ...
1
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3answers
122 views

Proof of the Principle of mathematical Induction [duplicate]

We always use the Principe of Mathematical induction and we have two versions of it. I myself have been using it for many years. But it just came to my mind that I have never seen a proof of the ...
3
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1answer
47 views

Induction proof for a summation: $\sum_{i=1}^n i^3 = \left[\sum_{i=1}^n i\right]^2$ [duplicate]

Prove by induction: $\sum_{i=1}^n i^3 = \left[\sum_{i=1}^n i\right]^2$. Hint: Use $k(k+1)^2 = 2(k+1)\sum i$. Basis: $n = 1$ $\sum_{i=1}^1 i^3 = \left[\sum_{i=1}^1 i\right]^2 \to 1^3 = 1^2 \to 1 = 1$. ...
1
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2answers
46 views

Proof by Induction that $16 \mid 5^n - 4n - 1$

Using induction, prove that $16\mid 5^n - 4n - 1$ for $n$ in $\mathbb{N}$ Here's what I have and what I'm stuck on: basis: $n = 1$, $5 - 4(1) - 1 = 0$ and $16\mid 0$. Hypothesis: Assume true for ...
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0answers
15 views

What method to use to find a hypothesis of the solution of the recurrence relation?

Suppose that we want to find an asymptotic upper bound for a recurrence relation: $T(n)=aT \left ( \frac{n}{b}\right)+f(n)$ , $T(n)=c, \text{ when } n \leq n_0$, using the following method: We choose ...
3
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1answer
344 views

Using induction to prove an equality in harmonic numbers

Question: Prove that harmonic numbers satisfy the equality using induction $$ H_{1}+ H_{2} + · · · + H_{n} = (n + 1)H_{n} − n. $$ I have done the basis step: $(1 + 1)H_{1} − 1 = 1$. Correct. Done the ...
0
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1answer
44 views

True or Flawed proof

Is the following proof correct or flawed? (a) Claim: For every positive integer $n, n^2 + 3n$ is odd. Proof: The proof will be by induction on $n$. Base Case: The number $n = 1$ is odd. Induction ...
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2answers
43 views

Prove (by induction?): If $A \subset \mathbb{N}$, $4 \in A$ and $n+1 \in A$ whenever $n \in A$, then $\left\{n \mid n \geq 4 \right\} \subset A$.

Prove: If $A \subset \mathbb{N}$, $4 \in A$ and $n+1 \in A$ whenever $n \in A$, then $\left\{n \mid n \geq 4 \right\} \subset A$. So for the base case, I did $n = 4$, so we have $4 \in A$ by ...
0
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1answer
55 views

Prove by induction that $7^n < n!$ for all integer $n \ge 21$ [closed]

Prove by induction that $7^n < n!\,$ for all integers $n\ge 21$
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2answers
187 views

Proof by induction failure if assumption is wrong?

I never got a clear answer to this question in college. What happens in an induction proof if the assumption is wrong? For example, suppose we try to prove that $n^5$ > n! for n >= $2$ so we start ...
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4answers
38 views

Prove $(n!)/n^n \leq 1/2^{k}$, where $k$ is the floor of $n/2$.

I suppose the natural way to prove this is by induction. When I follow the rather natural steps $$\frac{(n+1)!}{(n+1)^{n+1}} = \frac{n!}{(n+1)^{n}} \leq \frac{n!}{(n)^{n}}$$ in order to apply the ...
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1answer
43 views

Proof of series with induction

I have the sum ...
0
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1answer
40 views

How to prove the exponent law with rational exponents by Induction

May I know how to prove that $b^n \times b^m = b^{n+m}$ given that the exponents are now rational numbers instead of pure integers ?
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1answer
28 views

Prove by induction that $\sum_{i = 1}^{n} \frac{1}{\sqrt{i}} \leq 2\sqrt{n} - 1$

Prove by induction that $\sum_{i = 1}^{n} \frac{1}{\sqrt{i}} \leq 2\sqrt{n} - 1$ I want to do the $n - 1 \rightarrow n$ induction step. But I'm confused as to what my base case is. Usually if I want ...
0
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3answers
44 views

Prove that $S_n = 5^n - 1$

Use Strong Induction: $s_0 = 0 $, $s_1 =4$ and $s_n= 6s_{n-1} - 5s_{n-2}$ for all $n\in \mathbb{N} \setminus \{1\}$ Prove that $S_n = 5^n - 1$ In regards to the first step, can I start at n=2? Not ...
0
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1answer
34 views

Prove by mathematical induction

I stuck with a problem like this. I know all the steps but I can't prove that it is true when n=k+1. n^2 ≥ 2n + 1, for all n ∈ N such that n ≥ 3.
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6answers
167 views

Prove that if $n^2$ is divided by 3, then also $n$ can also be divided by 3.

$n\in \Bbb N$ Prove that if $n^2$ is divided by 3, then also n can also be divided by 3. I started solving this by induction, but I'm not sure that I'm going in the right direction, any ...
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3answers
35 views

Proving inequalities using induction all natural numbers that's greater than or equal to 5

using mathematical induction, prove that $n\le5: 4n<2^n$ base case: $4(5) < 2^5$ $20 < 32$ Correct I need help with the inductive process
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4answers
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How can I show that $n! \leqslant (\frac{n+1}{2})^n$?

Show that $$n! \leqslant (\frac{n+1}{2})^n \quad \hbox{for all } n \in \mathbb{N}$$ I know that it can be done by induction but I always find line where I do not know what to do next.
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2answers
36 views

Proof by induction with variable other than $n$ [duplicate]

1) Prove that $(1+x)^{n} \geq 1 + nx$ for every $n \in \mathbb{N}$ and $x \in (-1, \infty)$ Base case: Usually for the base case I just take $n = 1$ but since there's another variable $x$, I wasn't ...
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2answers
43 views

Prove by induction that $r_0 + r_1a + r_2a^2 + \cdots + r_{n−1}a^{n−1} < a^n$.

Let $a$ be a natural number greater than $1$. Prove that for all integers $r_0 , r_1 , \cdots , r_{n−1}$ with $0 ≤ r_j < a$, we have: $$ r_0 + r_1a + r_2a^2 + \cdots + r_{n−1}a^{n−1} < a^n ...
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2answers
508 views

Proving that if one person in any group of four knows three, then someone knows everyone.

title can't exactly capture the description of this problem so well. Here's the question in full: "At a convention, any group of four people contains one who knows the other three. Prove there is ...
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4answers
43 views

Use Induction to prove: $(1+2x)^n \geq 1+2nx$

Show by induction that: for all $x>0$ that $(1+2x)^n \geq 1+2nx$ So far I have: for $n=1 \rightarrow (1+2x)^1 \geq 1+2x$. True! for $n=k+1 \rightarrow (1+2x)^{k+1} \geq 1+2(k+1)x$ ...
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1answer
46 views

characteristic function differentiation

Let $\mu$ be a probability measure on $\mathbb{R}$. Then the characteristic function is: $$ \varphi: \mathbb{R} \rightarrow \mathbb{C} \;\;\ \varphi(t):=i\int_\mathbb{R} e^{itx}d\mu(x) $$ Prove with ...
2
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2answers
128 views

$\binom{n} {0} F_0+\binom{n}{1} F_1+\binom{n}{2} F_2+\cdots +\binom{n}{n} F_n=F_{2n}$

Please help! I need help on my assignment for discrete mathematics! Prove the following identity: $\binom{n} {0} F_0+\binom{n}{1} F_1+\binom{n}{2} F_2+\cdots +\binom{n}{n} F_n=F_{2n}$ I need to ...
2
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2answers
64 views

Prove by mathematical induction that exponentials grow faster than polynomials

How to prove that for $\forall q>1$ $\forall k\in \mathbb{N}$ $\exists c>0$ $\forall \in \mathbb{N}$ $q^n≥cn^k$? I should use mathematical induction.
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2answers
37 views

Combinatorics identity proof by induction

Prove the formula by induction on n and fixed r: $\binom{r}{r} + \binom{r+1}{r} + \binom{r+2}{r} + \ldots + \binom{n}{r} = \binom{n+1}{r+1}$ What I tried: Base: we take $n=r$ so $\binom{r}{r} = ...
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2answers
41 views

Inductive proof of the degree of a polynomial

Here is the problem: Assume that there is a polynomial $P(x)$ of degree 4 such that for all $N \in \mathbb{N}$, $$P(N) = \sum\limits_{n=0}^N n^3$$ Find the polynomial. Use induction to prove that ...
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2answers
81 views

Function composition: $f^{653}(56)=?$

Let $f(x) = \frac1{(1-x)}$. Define the function $f^r$ to be $f^r(x) = f(f(f(...f(f(x)))))$. Find $f^{653}(56)$. What I've done: I started with r=1,2,3 and noticed the following pattern: $$f^r(x)= ...
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2answers
48 views

Prove By Induction (Fibonacci Sequence)

Prove by PMI $\gcd(f_n,f_{n+1}) = 1$ for all natural numbers $n$. $f_n$ represents the Fibonacci sequence.
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239 views

Number of particles at time $t$

A following problem appears in my text book under the section of induction: At time $0$, a particle resides at the point $0$ on the real line. Within $1$ second, it divides into $2$ particles that ...
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1answer
178 views

prove weak induction implies strong induction

There is a solution from a year ago that I don't quite follow which is why I post this along with my attempt, so it is not a duplicate. Prove weak induction implies strong induction: weak ind. ...
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0answers
39 views

Game of writing a binary sequence proof

Let $n \gt 2$ be a natural number. We consider the following game. Two players write a sequence of $0$s and $1$s. They start with an empty line and alternate their moves. In each move, a player writes ...
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1answer
87 views

Using induction for $x^n - 1$ is divisible by $x - 1$

Prove using induction that for all non-negative integers n and for all integers $ x > 1 $, $ x^n - 1 $ is divisible by $ x - 1 $. Step 1: We will prove this using induction on n. Step 2: Assume ...
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3answers
53 views

Proof that when repeatedly splitting a heap of marbles into two and writing down the product of the two heap sizes, the total is ${x \choose 2}$

Here is the problem in full: A heap has $x$ marbles, where $x$ is a positive integer. The following process is repeated until the heap is broken down into single marbles: choose a heap with more ...
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3answers
49 views

Proof by Induction: Series of binomial coefficients with same k-length subsets

I have no idea how to prove this binomial equation identity. For reference this is included in Discrete Mathematics for Computer Scientists by Clifford Stein, Robert Drysdale and Kenneth Boggart, ...