Mathematical induction Definition and 215 Threads

I know that this can be proven with combinatorial proof. But, here I want to prove by induction. I fix ##m,n## and try to induction on ##p##. Let ##P(m,n,p)## be the statement which needs to be proven. Base case is ##p=0##. LHS (left hand side) becomes $$ \sum_{k=0}^0...

If ##p## is even and ##q## is odd, then ##p = 2m## and ##q = 2n - 1##, for some ##m, n \in \mathbb{N}##. Let ##P(m,n)## be the statement that ## \binom{2m}{2n-1}## is even. Now, I am going to use mathematical induction here. My plan is to prove ##P(1, n)## and ##P(m, 1)## as base cases. For the...

I have already shown base case above for ##n=2##. Let ##k \geq 2## be some arbitrary in ##\mathbb{N}##. Suppose the statement is true for ##k##. So, this means that, number of k-digit binary numbers that have no consecutive 1's is the Fibonacci number ##F_{k+2}##. And I have to prove that...

Here is my attempt. Let ##P(n)## be the statement $$ \sum_{k=0}^{n} \binom{k}{r} = \binom{n+1}{r+1} $$ where ##1 \leq r \leq n##. Now base case is ##n = 1##. For this, we have $$ \sum_{k=0}^{1} \binom{k}{r} = \binom{0}{r} + \binom{1}{r} $$ Since ##1 \leq r \leq 1##. It means that ##r =...

Following is my proof. Let ##P(n)## be the statement that ##5 | (2^n a) \rightarrow (5 | a)##. Here ## n \in \mathbb{N}## and ##a \in \mathbb{Z}##. Base case is ##n = 1##. Suppose that ##5 | (2^1 a)##. This means that ## 2a = 5m## for some ##m \in \mathbb{Z}##. Now ##5m## is even. Since ##5## is...

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...

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...

I'm in need of assistance for the following attachement.

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...

$S_k>k^2$ $S_{k+1}:2^{k+1}>(k+1)^2$ $2*2^{k+1}>2(k+1)^2$ $2^{k+2}>2(k+1)^2$ Assume $x=k+1$ $\frac{2^{x+1}}{2}>x^2$ $2^{x+1}*2^{-1}>x^2$ $2^x>x^2$ right? $2^{k+1}>(k+1)^2$

$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...

Are these satisfactory answers? Question 1: http://i.imgur.com/EElOqwM.jpg Question 2: http://i.imgur.com/SlE2jza.jpg?1 Thank you :)

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...