What is Mathematical induction: Definition and 213 Discussions
Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, . . . ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3), . . . . Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder:
Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step).
A proof by induction consists of two cases. The first, the base case (or basis), proves the statement for n = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that if the statement holds for any given case n = k, then it must also hold for the next case n = k + 1. These two steps establish that the statement holds for every natural number n. The base case does not necessarily begin with n = 0, but often with n = 1, and possibly with any fixed natural number n = N, establishing the truth of the statement for all natural numbers n ≥ N.
The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction is an inference rule used in formal proofs, and in some form is the foundation of all correctness proofs for computer programs.Although its name may suggest otherwise, mathematical induction should not be confused with inductive reasoning as used in philosophy (see Problem of induction). The mathematical method examines infinitely many cases to prove a general statement, but does so by a finite chain of deductive reasoning involving the variable n, which can take infinitely many values.
The Student's Manual simply applies l'Hospital's Rule n-times to arrive at ##\frac{e^x}{n!}\to\infty## as ##x\to\infty##.
However, I'm wondering if I could use Mathematical Induction to prove this. Is the following correct and sufficiently rigorous (at least for an undergraduate-level Calculus...
Ok, would i be correct to approach it this way,
Let ##n=1##. If ##n=1##, then ##5^1+3## is divisible by ##4##, the statement is true for ##n=1##.
Assume its true for ##n=k## ∀ ##kε\mathbb{z}^{+}.## Then ##5^k+3## is divisible by ##4.##
i.e ##5^k+3=4m## ∀ ##m ε\mathbb{z}^{+}##
Let ##n=k+1.##...
I have gone through the principle of mathematical induction. I cannot understand why do we need to prove every statement for n=1. I mean why is it necessary? Why can't we start directly from n=k then n=k+1. For example see the below image. Thanks!
Summary:: prove that (n 0) + (n 1) + (n 2) + ... + (n n) = 2^n is true using mathematical induction.
note that (n n) is a falling factorial
Hello! I have trouble dealing with this problem:
Mod note: Thread moved from math technical section, so is missing the homework template.
Prove that (n...
Prove by mathematical induction that 7^n-2^n is divisible by 5.What I've done so far:For n = 1
7^1-2^1=7-2=5 (true that it is divisible by 5)
For n = k
7^k-2^k=5a (assumed to be true that it is divisible by 5)
For n = k + 1...
I ran into this kind of expression for a sum that appears in the theory of 1-dimensional Ising spin chains
##\displaystyle\sum\limits_{m=0}^{N-1}\frac{2(N-1)!}{(N-m-1)!m!}e^{-J(2m-N+1)/kT} = \frac{2e^{2J/kT-J(1-N)/kT}\left(e^{-2J/kT}(1+e^{2J/kT})\right)^N}{1+e^{2J/kT}}##
where the ##k## is the...
Homework Statement
[/B]
Weak Induction:
If (i) ##S(1)## holds, and (ii) for every ##k \geq 1(S(k) \Rightarrow S(k+1)##. Then ##\forall n \geq 1##, ##S(n)## holds.
Strong Induction:
If (i) ##S(k)## is true and (ii) ##\forall m\geq k [S(k) \land \cdots \land S(m)]\Rightarrow S(m+1)##. Then for...
On the outside rim of a circular disk the integers from 1 through 30 are painted in
random order. Show that no matter what this order is, there must be three successive
integers whose sum is at least 45.
Dear all
I am trying to prove by induction the following:
I checked it for n=1, it is valid. Then I assume it is correct for some k, and wish to prove it for k+1, got stuck with the algebra. Can you kindly assist ?
Thank you.
Homework Statement
Prove that 1\cdot1!+2\cdot2!+...+n\cdot n! = (n+1)!-1 whenever n is a positive integer.
Homework EquationsThe Attempt at a Solution
I'm having trouble simplifying towards the end of the proof.
Proof:
Let P(n) be the statement 1\cdot1!+2\cdot2!+...+n\cdot n! = (n+1)!-1...
Homework Statement
Wherein α, β are strings, λ = ∅ = empty string, βr is the shortest suffix of the string β, βl is the longest prefix of the string β, and T* is the set of all strings in the Alphabet T, |α| denotes the length of a string α, and the operator ⋅ (dot) denotes concatenation of...
Homework Statement
Wherein α is a string, λ = ∅ = the empty string, and T* is the set of all strings in the Alphabet T.
Homework Equations
(exp-Recursive-Clause 1) : α0 = λ
(exp-Recursive-Clause 2) : αn+1 = (αn) ⋅ α
The Attempt at a Solution
[/B]
This one is proving difficult for me. I...
Homework Statement
Use mathematical induction to prove that (8n − 7n − 1) is divisible by 49 for any n ∈ N.
Correction by mentor for better readability: ##49\,|\,(8^n-7n-1)##
The Attempt at a Solution
We can see that the base case is satisfied here:
n = 1,
8^1-7*1-1 = 0 and 49 | 0 is true...
Prove by mathematical induction that
n
sigma r^3 = n^2(n+1)^2/4
r = 1
so far I have
1
sigma r^3 = 1^2(1+1)^2/2
r=1
1 = 1(4)/2
1 = 4/2
1 = 2
I'm not sure what to do after this for the k+1 case.
<Moderator's note: Moved from a technical forum and thus no template.>
A^n = 1 2^n
0 1
Prove your formula by mathematical induction.
I began by taking A to successive powers but not sure of what my formula should be.
A^0 = 1 0 , A^1 = 1 2 , A^2 = 1 4 , A^3 = 1 6 ...
Homework Statement
Prove ##5^n+9<6^n## for ##n\epsilon \mathbb{N}|n\ge2## by induction.
Homework Equations
None
The Attempt at a Solution
The base case which is when ##n=2##:
##5^2+9<6^2##
##34<36##
Thus, the base case is true. Now for the induction step.
Induction hypothesis: Assume...
In mathematical induction we prove statements using a proper technique just like in a following example:
##P(n)=n(n+1)## is an even number for every ##n∈N##
we will check wether ##P(1)## is True/false, its true because ##P(1)=2##
Now We will assume that ##P(k)## is true and using this we will...
Given a set, there are subsets and possible relations between those arbitrary subsets. For a given example set, the possible relation between the subsets of the example set will narrow down to the "true" possible relations between those subsets.
a) {1}
Number of Subsets: ##2^1 = 2## (∅, {1})...
I'm still learning English, had to use dictionary and translator, so I'm sorry if its unclear, i will try to explain it more if needed.
Homework Statement
For n belonging to N when n is even and n > 3, prove that
(4^(n-3) + 5^(n-3) + 9) is divisible by 9
Homework Equations
3. The Attempt at...
Homework Statement
Show that every 2-tree(Definition 3.2) with n internal nodes has n+1 external nodes
Homework Equations
A 2 tree is a tree which nodes have 2 children or no children.
Internal nodes are nodes which have two children
external nodes are leafs.
The Attempt at a Solution
Proof...
Homework Statement
Prove that \forall n\in \mathbb{N}
\sum\limits_{k=1}^{n}(-1)^{k+1}{n\choose k}\frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}
Homework Equations
-Induction
-Summation
-Binomial coefficient
The Attempt at a Solution
For n=1 equality is true.
For n=m
m-{m\choose...
Prove that n^5 - 5n^3 + 4n is divisible by 120. for every natural number n greater or equal to 3.
First, i checked if it works for n=3 and it does,
so i could assume it works for some k>=3 so i could write k^5 - 5k^3 + 4k as 120*a a is natural number
so for k+1 i have:
(k+1)^5 - 5(k+1)^3...
Hi,
I've seen on on several sites that you can prove that nCr, where r<=n, is a natural number. I'm not sure how to do this by induction.
So I need help on this proof. How do I write this as a mathematical statement at the start of the induction proof?
Thank you
Homework Statement
Prove that (n2+3)(n2+15) is divisible by 32 for all odd positive integers n.
Homework Equations
I suppose we are supposed to use mathematical induction since it is in that chapter, but the following questions specifically state that we should use induction but this question...
Homework Statement
A step in this process of proving Sn: 1+4+7+...+(3n-2) = n(3n-1)/2
confuses me. I hope someone can clarify this for me.
I do not require the work done, I need clarification on a step only. Thanks!
Homework Equations
After assuming n=k, we say Sk: 1+4+7+...+(3k-2) =...
Homework Statement
I am just learning the joys of mathematical induction, and this problem is giving me fits.Homework Equations
I am trying to prove that 2 + 4 + 6 + … + 2n = [2n(n+1)]/2The Attempt at a Solution
The base case is to prove P(1) is correct. Simple enough -- 2 = [2 x 1 (1+1)]/2...
$$
T(n) =
\begin{cases}
2 & \text{if } n = 2 \\
2T(\frac{n}{2})+n & \text{if } n = 2^k \text{, for } k > 1
\end{cases}\\ \text{ }
\\ \text{ }
\\ \text{ }
\\
\text{Prove } T(n) = n\lg(n) \text{ if } n = 2^k\text{, for } k > 1.$$
I am crawling through the "Introduction to Algorithms" textbook...
I am confused on a math problem. I am supposed to use mathematical induction to show that
(Summation with n on top and i=1 on the bottom) (2i-1) = 1+3+5+...+(2n-1)= n^2
Ok so i got step (a) and found that $\int_{0}^{\infty} \,d (x^0)*e^(-ax)dx=1/a$
But i do not get how i should go about starting the next steps using the info from the first step(have not done a similar problem before so i need to get a grasp on the method)
Hello,
I need to prove the following:
\sum_{i=0}^n\binom{n}{i} = 2^n
by using something called mathematical induction. I understand, somewhat, what it is - we propose a statement and show that is true for n=1, then we assume that the statement is true for all n \in \mathbb{N}, which should also...
I'm not sure if I'm posting in the right subforum because there is one about proofs, but it requires the question not to be homework-like, but I also need an explanation...
Homework Statement
Prove by mathematical induction:
Let m be the smallest element of A\subseteq\mathbb{N}. If A...
Can you help me with this exercise?
1^{1}+2^{2}+3^{3}+4^{4}+...+n^{n} = n^{n+1}
Thanks!
PD. I was trying to solve, and i have this:
1^{1}=1^{1+1} =
1 = 1
a) k^{k+1}
b) k+1^{k+1} = k+1^{(k+1)+1}
a in b) k^{k+1} + k+1^{k+1} = k+1^{(k+1)+1}...
Homework Statement
Find the error in the following proof that \all horses are the same color" 4.
Claim: In any set of h horses, all horses are the same color.
Proof: By induction on ##h##:
Base Case: For ##h = 1##. In any set containing just one horse, all horses are clearly the same color...
Hello,
We had a short unit on mathematical induction, and I know my final exam will probably have one problem that says ''prove this relationship with mathematical induction''. I was wondering, is there some sort of step by step procedure to proving something using induction? Or is it...
$S_k:5\cdot 6 +5\cdot 6^2+5\cdot 6^3+ ...+5\cdot 6^k=6(6^k-1)$$S_k:5\cdot 6 +5\cdot 6^2+5\cdot 6^3+ ...+5\cdot 6^k+ 5\cdot 6^{k+1}=6(6^k-1)+5\cdot 6^{k+1}$
what do i do now? to prove $S_{k+1}$
Definition/Summary
Mathematical Induction is a method of proving a series of mathematical statement labelled by natural numbers.
This method usually involves two steps. First one proves the base case, then one shows that if the statement holds for some natural number, it holds for the...
Did i do this right?
$2^n>n^2$, $n \ge 5$
$S_{k+1}: 2^{k+1}>k^2+2k+1$
$2^k>k^2$
$2(2^k)>2k^2$
$2^{k+1}>k^2+k^2$
And $k^2+k^2>k^2+2k+1$
RHS: $k^2+2k+1$
So $2^{k+1}>(k+1)^2$
I don't know what forum to post this under. PLEASE HELP ME THOUGH!
Principle of Mathematical induction:
Let $S_n$ be a statement concerning the positive integer n. Suppose that,
$S_n$ is true.
For any positive integer k, k $\le$ n, if $S_k$ is true, then $S_{k+1}$ is also true.
Then $S_n$ is...
Original Equation: 5+6+7+...+(n+4)=1/2n(n+9)
Ok, I've tried everything to understand this. I'm just not getting it. I understand everything (n=1, k+1, etc) up until this point: "To continue with proof what must be done?". I know you must simplify the right side, but I don't understand how they...
Homework Statement
Homework Equations
The Attempt at a Solution
Hello, I am working on the problem in the attached image regarding induction based on the inequality
2^n \geq n + 1
I am confused how to do this by proving that if it is assumed to be true for ## n = k ##, then how it is true...
Homework Statement
(5^2) + (11^2) + (17^2) +...+ (18n-1)^2
a)Write the sum in sigma notation
b)Using the sigma identities solve the sum (easy to do)Homework Equations
∑i = .5*k*(k+1) etc
The Attempt at a Solution
The problem I'm having is with the 25 and 121. I thought it was
∑ (6n-1)^2...
hi i have this homework question and I am not sure if my thought process is valid.
The Question:
let a, b and c be roots of the polynomial equation: x^3+px+q=0 and S(n)=(a^n)+(b^n)+(c^n)
now prove: that for S(n)= -p(S(n-2))-q(S(n-3)) for n>3my attempt:
-------------
first off...
Recall that the fibonacci sequence is defined as
{ f0=0; f1 = 1 and
{fn = f n - 1 + fn -2 for n 2
Prove by generalized mathematical induction that
fn = 1/sqrt(5)[ϕn - (-ϕ)-n]
where ϕ = [1+sqrt(5)]/2
is the golden ratio.. (This is known as de Moivre's formula.)
So...
Hey!
I've just started to learn some mathematical induction and it's proving quite tricky. Hopefully someone could help me out a little with my questions!
I don't know how to format question on this forum but I'll try my best to accurately represent it:
n
sigma i * 2^i = (n-1) * (2^n+1) + 2...
With mathematical induction i should prove that is true for all natural numbers:http://img.tapatalk.com/d/13/10/18/u6epesu5.jpg
Im sorry beacuse i have inserted an image,but I am still not used to write it in [MATH] here...