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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What actually constitutes a *definition* for a function?

I'm reading Enderton's text on logic trying to justify ( to myself ) the use of induction on recursively defined sets. This is of course used repeatedly in trying to prove results about well-formed ...
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1answer
19 views

How do you solve a recurrence with a functin through induction?

I found the answer in part-A by substitution, as O(n) from; T(n/2^k) = T(1).... n/2^k = 1..... so k = 1og2(n)..... T(log2(n)) = T(n/n)+5.... so O(n) IS THE ANSWER, Correct me if am wrong because am ...
2
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5answers
119 views

Showing the $n$-th derivative of $\cos x$ by induction

I was asked to show that the $n$-th derivative of $\cos x$ is $\cos(\frac{n\pi}{2} + x)$. My progress : By induction, I proved it was true for $n=1$. Then I assumed it was true for $n = k$ so now I ...
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2answers
104 views

Proof that $\dfrac{1}{e^x}=e^{-x}$ without converting it to $e^{x}e^{-x}=1$.

I want to show that $\dfrac{1}{e^x} = e^{-x}$ from the Taylor expansion of $e^x$. To express $\dfrac{1}{e^x}$ as a power series, I let: $$ \left(\dfrac{1}{0!}x^0 + \dfrac{1}{1!}x^1 + ...
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4answers
156 views

Proof by induction (exponents)

Use proof by induction and show that the formula holds for all positive integers:$$1+3+3^2+\dots+3^{n-1}=\frac{3^n -1}2$$ The confusing step in my opinion is the first expression: $3^{n-1}$, when ...
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1answer
52 views

Show that $f^{[n]}(0)=0$ for all $n=0,1,2…$

Let $$f(x)=\left\{ {\matrix{ {{e^{ - {1 \over {{x^2}}}}},x \ne 0} \cr {0,x = 0} \cr } } \right.$$ Show that $f^{[n]}(0)=0$ for all $n=0,1,2,\cdots$ The proof: First note that for ...
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2answers
72 views

Induction Proof of trig inequality

This is for a course, I don't want the answer just a prod in the right direction! I've got a problem that states let n be an integer such that $$n\gt0$$ $$\text{Prove: }\sum_{k=0}^n |\cos k| \ge ...
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1answer
42 views

Use the recursive definition, $f(k) + k² + 2k -3$, to prove that, for any $n ∈ \Bbb N,\ f(n) = n² - 4$.

Here's what I have so far: Let $f: \mathbb{N} - \{1\} \to \mathbb{N}$, such that $f(x) = (x+2)(x-2)$. Formulate a recursive definition for $f$ including both the base case $f(2) = 0$ and a ...
2
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2answers
105 views

coin problem with two coins, inductive proof

Adjustment This proof is flawed. I want to ask something about the coin problem with two coins. Let $a,b$ be to numbers in $\mathbb{N} \setminus \{0\}$ (elsewhere I include zero) which have no prime ...
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2answers
70 views

Finding a formula for $1+\sum_{j=1}^n(j!)\cdot j$ using induction

I need help with finding the formula and proving it by induction. Am stuck, but the professor says we should know this by now.
3
votes
1answer
40 views

Direct proof of principle of transfinite induction

This is a problem from the book Set theory by You-Feng Lin. Principle of Transfinite Induction Let $(A,\le)$ be a well-ordered set. For each $x \in A$, let $p(x)$ be a statement about $x$. If for ...
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1answer
63 views

Prove $1! + 2! + . . . + n! < (n + 1)!$ using mathematical induction [duplicate]

$1! + 2! + . . . + n! < (n + 1)!$ This question has left me stumped for quite some time. I am not sure how to approach it. (I am really bad at induction).
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3answers
48 views

how to prove $1/n (1-(1/2)^n)$ decreasing without using differentiation

$a(n)=1/n (1-(1/2)^n)$ prove $a(n+1)<a(n)$ for n>0 by differentiating slope comes negative and then we can prove it . but i wanted to solve it without that . can someone help
2
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3answers
43 views

How to solve this recursive equation?

I've got this recursive equation: $$ T(n) = \begin{cases} 2, & \text{if $n = 2$} \\ 2T(n/2) + n, & \text{if $n = 2^k$ where k > 1, $k \in \mathbb{N} $} \\ \end{cases} $$ I know I should ...
2
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1answer
114 views

Use induction to prove the following: $1! + 2! + … + n! \le (n + 1)!$

Use induction to prove the following: $1! + 2! + .... + n! < (n + 1)!$ Base case: $n = 1$ $1! < 2!$ true Inductive step: Assume that $1! + 2! + .... + k! \le (k + 1)!$ is true let $n = k ...
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0answers
43 views

IS this proof by induction of the hand shake lemma correct?

Proof by induction that the sum of degrees of vertexes in an undirected graph equals two times the number edges, where $V$ is the set of vertexes and $E$ is an edge multiset: $$\sum_{v ∈ V} deg(v) = ...
1
vote
1answer
55 views

What is wrong with the following induction argument?

I found a problem on a note on induction. The problem went like this: "Let $n$ be a non-negative integer. Suppose we are given a triangle and n points inside it, with no three of the given $n + 3$ ...
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votes
5answers
191 views

The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer

Prove by induction that this number is an integer: $$u_n=(3+\sqrt{5})^n+(3-\sqrt{5})^n$$ Progress I assumed that it holds for $n$ and I tried to do it for $n+1$ but the algebra gets quite messy and ...
2
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1answer
47 views

Reference for $F$-algebras and induction?

I've been learning about $F$-coalgebras and coinduction from this fantastic paper, which has really helped me get a feel with its many examples. I'm starting to struggle with reconciling the ...
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2answers
49 views

Proof of the inequality $F_i<(5/3)^i$ for the Fibonacci numbers

The example states: As an example, we prove that the Fibonacci numbers, F0 = 1, F1 = 1, F2 = 2, F3 = 3, F4 = 5,..., Fi = Fi - 1 + Fi - 2, satisfy Fi < (5/3)i, for all i >= 1. To do this, we ...
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1answer
36 views

Induction Problem Number of Tiles on Floor

I took a discrete math course about a year ago, and I recently decided to crack open my book again as a refresher on induction proofs and problems. I ran across this problem, which I didn't remember ...
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1answer
66 views

Proof by Induction [Number Theory by George E. Andrews 1-1 #2] [duplicate]

I am to use mathematical induction to prove that: $$1^3 + 2^3 + 3^3 + \cdots + n^3 = (1 + 2 + 3 + \cdots + n)^2 $$
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3answers
64 views

Showing that $\sum\limits_{k=2}^n {k\choose2} = {{n+1}\choose 3}$ for integers $n\geq 2$

I'm trying to prove that $\sum\limits_{k=2}^n {k\choose2} = {{n+1}\choose 3}$ for integers $n\geq 2$. I figured induction was the way to go, so I tried. This is what I've accomplished so far: Proved ...
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1answer
71 views

Maximum and average number of inversions in array by induction

Just for your information, an inversion in an array $a$ is any ordered pair of points $(i, j)$ where $i < j$ and $a_i > a_j$. I can prove the maximum and average number of inversions in an ...
33
votes
5answers
4k views

Prove that the 25 people can be seated in this way

5 mathematicians, 5 biologists, 5 chemists, 5 physicists, and 5 economists sit around a large round table. Prove that the 25 people can be seated such that, if A and B are two different people with ...
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votes
1answer
60 views

Prove if n<m there is at least one [(n/m)]?

If there is n persons and m places to seat. Is it right that at least one seat which contain [(n/m)] persons? Note: I was not able to find the right sign [ is returning first upper integer in case of ...
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votes
2answers
85 views

Limit of a sequence of averages (three variables)

Let $a_0 = 0$, $a_1 = 0$, $a_2=1$ and for $n>2$, $a_n = \dfrac{a_{n-1}+a_{n-2}+a_{n-3}}{3}$. Consider $\lim\limits_{n \to +\infty} a_n$. Using a python script I found that $a_n$ tends to ...
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1answer
37 views

For every $n$ there exists $m$ such that $m/n$ is an upper bound but $(m-1)/n$ is not

This is a problem discussed in Analysis 1 by Terence Tao. $E$ is a non-empty set of Real numbers. $ n\geq 1$, $L,K$ are two integers such that $L<K$. Let $\frac{L}{n}$ is not an upper bound of ...
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4answers
85 views

Prove that $133$ divides $11^{n+1} + 12^{2n−1}$ for all $n > 1$? [duplicate]

I have tried this question so hard but still stuck here. It seems like easily provable if all $n$ are all positive numbers but in this question, the $n$ is bigger than $1$. original question : prove ...
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1answer
65 views

Induction on prime numbers

To dive straight into the question: is there a form of induction which works on prime numbers? I've thought, and while I'm pretty sure it can be done om numbers such as even numbers or numbers ...
2
votes
2answers
71 views

Show that $(n+1)^{n+1}>(n+2)^n$ for all positive integers

Show that: $(n+1)^{n+1}>(n+2)^n$ holds for all positive integers I tried using induction: for $n=1$ we have 4>3 then for $n+1$ we have to show that $(n+2)^{n+2}>(n+3)^{n+1}$ and here I ...
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1answer
33 views

how to prove using induction that sum of terms?

Prove that $\displaystyle\sum\limits_{i=1}^{k}\left(\dfrac{1}{(2i-1)}\dfrac{1}{(2i+1)}\right) = \dfrac{k}{(2k+1)}‎‎$ My Base of Induction is to check that it is true for i=1, so: ...
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1answer
56 views

Proof of equation $\sum_{k}{n\brack k}a_k = n!2^{n-1}$ by induction

I'm trying to prove to following equation: $$\sum_{k=0}^{n}{n\brack k} a_k = n!2^{n-1};\ \ \ n\ge 1$$ $a_n$ - number of ordered partition of set. We have following recursion dependencies: $a_n = ...
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1answer
34 views

Readings on more general/abstract notions of induction related to logic

Can someone suggest references to understand the more general/abstract concept of induction? Specifically, I am trying to justify to myself what is called induction on the "complexity of a ...
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1answer
37 views

Induction proof for continued fractions

Recently while preparing for a maths test, I got this question in a book: Let $a(n) = 3 + \cfrac{1}{3+\cfrac{1}{3+\cfrac{1}{3+\cdots }}}$ till $n$ terms. Prove that $a(n) \cdot a(n-1)=3a(n-1)+1$ ...
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What's wrong with this induction based proof?

Claim: $\forall x \in \mathbb{R^+} ,$ $ x^n=1 $ $where$ $ n\in \mathbb{N}$ Proof by induction on n: Basis step: $\forall x \in \mathbb{R^+} ,$ $ x^0=1 $ Induction Step: Let this holds for all ...
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2answers
26 views

Help me find the wrong in this inductive method proof?

Problem: Prove that: In a classroom with n student, if there is a girl student, all students of this class are girl. Solving: Let f(n) is the clause: In the class, if there is 1 girl student, all of ...
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1answer
71 views

Equivalence between “mathematical induction” and “transfinite induction” for natural numbers?

The "principle of mathematical induction" says that for a subset $S$ of $\omega$ (where $\omega$ is the set of all natural numbers), if $0 \in S$ and $n \in S \implies n^+ \in S$, then $S = \omega$. ...
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Proof Strategy: Induction Summation of Series

Let $P(n)$ be the following statement: $$\sum\limits^{n}_{i=0}r^i = \dfrac{1-r^{1+n}}{1-r}\text{ for all }n \in \mathbb{N}\text{.}$$ I am stuck at the base case: $$P(1):1 + r = ...
2
votes
2answers
37 views

Proof by Induction: $(1+x)^n \le 1+(2^n-1)x$

I have to prove the following by induction: $$(1+x)^n \le 1+(2^n-1)x$$ for $n \ge 1$ and $0 \le x \le 1$. I start by showing that it's true for $n=1$ and assume it is true for one $n$. ...
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2answers
57 views

How to simplify the formula for $n$th Fibonacci number when $n=2$?

When n is equal to 2 how do I simplify when the $n=2$ is put into the equation below (by the way I have to prove this formula by induction that when n= any number it will equal that number) ...
3
votes
3answers
90 views

Proof by induction: $n$th Fibonacci number is at most $ 2^n$

I'm trying to find the proof by induction of the following claim: For all $n\in\mathbb N$, $\operatorname{fibonacci}(n) \le 2^n$ My Proof: Base case: $n = 1$ $\operatorname{fibonacci}(1) \le 2^ 1$ ...
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2answers
94 views

Proof of equality $\sum_{k=0}^{m}k^n = \sum_{k=0}^{n}k!{m+1\choose k+1} \left\{^n_k \right\} $ by induction

I have a problem with following equality: $$\sum_{k=0}^{m}k^n = \sum_{k=0}^{n}k!{m+1\choose k+1} \left\{^n_k \right\} $$ And I would like to use induction in following way: Base: $$ m = n $$ And: $$ ...
0
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2answers
37 views

How to use the Comparison Test to investigate the convergence of $\sum (\ln n)/n^\alpha$?

Let $$\sum\limits_{n=1}^\infty \frac{\ln n}{n^\alpha}, \alpha\in\Bbb{R}$$ I need to investigate the convergence of this series. I've read that since the series is positive for all $n$ then it ...
0
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2answers
56 views

Mathematical Induction - Inequality

Does anyone have any idea on how to complete the inductive step? Thm: For all $n >= 0~~~~ 6^n + 4 > n^3$ Pf: by Induction     Let $P(n)$ be proposition that $~6^n + 4 ...
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2answers
43 views

Prove by induction a formula for $x_{k+1}=\frac{x_k}{x_k+2}$, $x_1=1$

I have a IT Maths exam coming up and I just can't figure out this question. Any help would be appreciated thanks. A sequence of integers $x_1,x_2,\dots,x_k,\dots$ is defined recursively by ...
0
votes
1answer
31 views

induction exercise doubt

the exercise states: Let $x_1 , ...,x_n$ be strictly positive numbers such that their product is equal to 1. Show then that $\sum_{k=1}^{n} {x_k} \ge n $, for every $n \ge 2$. My solution: for the ...
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0answers
85 views

proof by mathematical induction that the total number of subsets of a set is $2^n$ [closed]

What is the proof by mathematical induction that the total number of subsets of a set is $2^n$?
0
votes
2answers
38 views

Proof in induction recrusive function

We've got the following function: $$f:N \rightarrow N$$ $$f(0) = 1$$ $$f(K+1) = (K+1)\times F(K)$$ How can I proof in induction the following: $$1\times f(1)+2\times f(2)+3\times f(3)+...+n\times ...
3
votes
1answer
86 views

Divisor function asymptotics

Define $\tau_{r}(n) = \sum_{d_1...d_r = n}1$. One exercise in a book on sieve theory asked for an elementary proof by induction of the fact that $$\sum_{n\le x}\tau_r(n) = \frac{1}{(r - 1)!}x(\ln ...