Mathematical induction Definition and 218 Threads

1. I don't understand how to prove this. for all n≥1, 10n - 1 is divisible by 9. 3. I've done the basis step. Now I'm on the inductive step. I'm using (10k+1-1)/9=1. I don't know where to go from there. Using algebra just gets me down to 10k+1= 10. And I really don't think that's...

Hello everybody, I am doing my reading lately to prepare for some exams to join a mathematics department. And I would very much like, if anyone could help, the solution (or a hint) to the following induction proof. " Show that n^3 + (n+1)^3 + (n+2)^3 is a multiple of 9 " :smile: I...

Hi everyone, I have been trying to do this induction question... but can't seem to understand how to do it. Prove by induction that, for all integers n is greater than or equal to 1, (n+1)(n+2)...(2n-1)2n = 2^n[1x3x...x(2n-1)] This is what someone else has given as the answer: For n...

Homework Statement Okay, so I'm going to be completely honest, I am really bad at math, and I have been struggling the past couple of weeks in my Quantitative Reasoning class. I am so lost. I don't know if it's my teacher's teaching method or what, but nothing is clicking for me at the moment...

Homework Statement I need to prove that for any real number r, if 0 < r < 1, then for all positive integers n and m, if n < m, then r^n > r^m. Homework Equations No calculus techniques are permitted, only mathematical induction. The Attempt at a Solution I know that any...

Hello, I have a question about inductive reasoning... Earlier this week my intro proofs class went over the logical structure of induction, and an example. The example was a proof of \Sigmai = n*(n + 1)/2 My main issue is the assumption that "p(k)" is true. What if it's not? I asked this in...

I'm having a bit of trouble understanding why the principle of induction is included as one of Peano's axioms. It seems like it should not be independent of the others. Obviously it can be stated as: If a predicate P is true only of natural numbers, P(0) is true, and also P(n)\rightarrow...

Hi. I am learning mathematical induction on my own and I came across this problem: Homework Statement Prove: 1*4 + 4*7 + 7*10 + ... + (3n - 2)(3n + 1) = n(n + 1)²2. The attempt at a solution Quick test for n=1: (3 -2)(3 + 1) = 1(1 + 1)² 4 = 4 Alright, so I rewrite this with, on the left...

Homework Statement Prove that \frac{d}{dz}z^n = nz^{n-1}\;\;\; \forall n\in\mathbb{N} using the Product Rule for differentiation and mathematical induction. Homework Equations \frac{d}{dz} f(z) = \lim_{\Delta z\to 0} \frac{f(z+\Delta z) - f(z)}{\Delta z} \frac{d}{dz}[f(z)g(z)] = f\,'(z)g(z) +...

Homework Statement Prove that 11^n - 4^n is a multiple of 7 Homework Equations N/AThe Attempt at a Solution I substituted k+1 in for n and simplified to get 11(11^k)-4(4^k) but after this point I get stuck. Any help would be appreciated.

Homework Statement Summation of i(i + 1) (with i going from i = 2 to i = n-1) = n(n-1)(n=1) / 3 a. Write P(2). Is P(2) true? b. Write P(k) c. Write P(k+1) d. Prove by mathematical induction that the formula holds true for all integers n \geq 2 Homework Equations...

I was working on my homework and I did two of the mathematical induction problems before this one and they were super easy but I must be forgetting something because I just can't seem to solve this one. 2+7+12+17...+(5n-3) = (n/2)(5n-1) So I know that step 1 is to prove that it works with...

so the problem is 3+7+11+15+...+(4n-1) = n(2n+1) so I know that step 1 is to plug in 1 for the right side and check that it equals three... 3=1(2(1)+1) and yes it equals 3 Then I know that you assume that 3+7+11+15+...+(4n-1) = n(2n+1) The next step is where I get confused I know that...

Homework Statement For each n\,\in\,\mathbb{N}, let p_n(x)\,\in\,\mathbb{Z}_2[x] be the polynomial 1\,+\,x\,+\,\cdots\,x^{n\,-\,1}\,+\,x^n Use mathematical induction to prove that p_n(x)\,\cdot\,p_n(x)\,=\,1\,+\,x^2\,+\,\cdots\,+x^{2n\,-\,2}\,+\,x^{2n}Homework Equations Induction steps...

Homework Statement Prove that (n + 1)n - 1 < nn for n ∈ Z+. [Hint: Induction is suggested. Write out the induction statement explicitly. Make one side of the inequality look like your induction hypothesis.] Homework Equations The Attempt at a Solution ^ That's what I have so far. I'm good...

Homework Statement Prove: 1+2(2+3+4+···+n)+(n+1)=(n+1)2-1 for all n within NHomework Equations PMI The Attempt at a Solution I tried applying PMI starting with the basis step, allowing n=1, assuming the equation would be true. However 1+2(1)+(1+1)=?=(1+1)2-1 yields 5=?=3 which is false. The...

Homework Statement Use mathematical induction to prove the following statement. n \geq 2,~ \sum_{k=1}^{n} \frac{1}{k^2}~<~1-\frac{1}{n} Homework Equations The Attempt at a Solution This is the third problem I've done of this type, so I am by no means an expert. So this attempt may be...

In the context of Set theory and Relations why do we use mathematical induction. Is there any deep relation between all these concepts or mathematical induction is only a separate concepts introduced in the textbooks after Sets and Relation ; Functions ; and then Mathematical induction.

Homework Statement The sequence a0 -> n is defined by ai = b+i*c. Prove by induction on n that the sum of the terms in the sequence is (n+1)(a0 + an)/2.Homework Equations The Attempt at a Solution I defined predicate P(n) as (n+1)(a0+an)/2. My goal is P(n+1), which is (n+2)(a0+an+1)/2. I...

Homework Statement Prove that \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}\geq\sqrt{n} for all n \in N Homework Equations The Attempt at a Solution p(1): \frac{1}{\sqrt{1}} = \frac{1}{1} = 1 = \sqrt{1} \geq \sqrt{1} Let \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} +...+...

Homework Statement Consider the sequence of real numbers x1, x2, x3,... defined by the relations x1 = 1 and xn+1 =\sqrt{1 + 2xn} 1. Use mathematical induction to show that xn+1 > xn for all n \geq 1. The Attempt at a Solution I'm a bit thrown off by this question because it...

Hey guys, this is a problem given to us by our professor in one of the worksheets. I would like an opinion to see if this proof is valid Homework Statement Use mathematical induction to prove that 3^{n}+7^{n}-2 is divisible by 8 for \foralln\inZ^{+}Homework Equations Base step: n=0...

Mathematical induction...please help me! _{}a1 =1, _{}a2 =1,_{}a3 =2,_{}a4 =3..._{}an = _{}an-1 + _{}an-2 is a Fibonacci series...Prove That ^{2}_{}an+1 = _{}an _{}an+2 =(-1)n

Im practicing inductions on my own and I am getting stuck on one in particular... S_n = \frac{3(3^n-1)}{2} for a_n = 3^n I know that S_1 = 3 when you plug 1 into the equation, because it is the first term in the sequence Therefore, S_1 = a_1 = 3 I need to prove then S_{n+1} =...

Hi, I'm trying to learn mathematical induction for proving inequalities, but there is just one step I cannot get past: finding another inequality that is added to the inductive hypothesis. For example, in this problem: Prove for all positive integers (n >= 1), prove 3^n + 2 >= 3n. I...

Homework Statement Let f(n + 1) = 3f(n) + 8, with f(1) = 11. Prove by induction that f(n) = 5 x 3^n - 4.Homework Equations The Attempt at a Solution I don't even know where to start! Any help would be appreciated. Thanks. :-)

Please help! Use mathematical induction to prove.

Homework Statement What is wrong with my solution?... I don't quite understand where do I go from there... Homework Equations The Attempt at a Solution

Homework Statement Explain the difference between The principle of Mathematical Induction and The principle of Strong Induction. One is easier than the other. Why? Homework Equations The Attempt at a Solution My book says that they are in fact identical. Proofs are almost the...

I'm trying to solve this problem from CH2 of spivak's calculus of which I am self-studying. Homework Statement Prove the following by mathematical induction: 1^3+...+n^3=(1+...+n)^2 Homework Equations To prove by mathematical induction, you test whether P(1) is true and if P(k) is...

Homework Statement Prove the Inequality for the indicated integer values of n. n!>2^n,n\geq4 Homework Equations The Attempt at a Solution For n=4 the formula is true because 4!>2^4 Assume the n=k k!>2^k Now I need to prove the equation for k+1 I can multiply both sides by 2 and have...

Homework Statement Use mathematical induction to prove the formula for every positive integer n. \sum_{i=1}^{5}i^5=\frac{n^2(n+1)^2(2n^2+2n-1)}{12} Homework Equations The Attempt at a Solution I know this will be true because the RHS is just your standard sums of powers of...

Is the principle of mathematical induction unable to prove theorem of negative numbers?

Can i use the strong form every time? Why do people still use the ordinary form?

I cannot get my head around this at all. Suppose that v is of type Vector of Integers and we know the following:- 1. v is sorted. 2. No two items in v are the same. 3. v.at(1) is 12. For an integer n, where 1 <= n <= v.size(), let p(n) be the following proposition: p(n): v.at(n) => 11...

Homework Statement All the terms of the arithmetic progression u1,u2,u3...,un are positive. Use mathematical induction to prove that, for n>= 2, n is an element of all positive integers, [ 1/ (u1u2) ] + [ 1/ (u2u3) ] + [ 1/ (u3u4) ] + ... + [ 1/ (un-1un) ] = ( n - 1 ) / ( u1un)...

Homework Statement prove: 0^2 + 1^2 + 2^2 + ... + n^2 = n(n+1)(2n+1)/6 Homework Equations The Attempt at a Solution I'm confused on how to prove this by induction. I'm not exactly sure what the goal of the rearrangement is after substituting (n+1). Any help is much...

Homework Statement Prove that for all the natural numbers n that 2^n > n Homework Equations The Attempt at a Solution Base Case is easy Then the inductive step you have 2^k > k as the inductive hypothesis show that p[k+1] holds 2^(k+1) > k+1 on the left side 2^(k+1) = 2^k * 2...

Hi I'm a high school student. I gave a proof for the following theorem, but I was told by some professors that this is trivial and using natural induction twice for the rationals will do the same thing. What do you think? Is it just redundant? Theorem: Let P(r) be a statement about r...

Homework Statement Prove that H1 +H2 +...+Hn = (n +1)(Hn-n)? Homework Equations Hn denotes the nth harmonic number. The nth harmonic number is the sum of 1+1/2+...1/n, which is n / n +1. I'm not really sure if Hn = (1/ n) . Prove by Mathematical Induction Hn denotes the...

(a) Give a recursive definition of the set P of all non negative integers, (b) formulate the applicable induction principle and (c) then apply the induction principle to prove that 1/2^0+1/2^1+1/2^2...+1/2^i = 2-1/2^n for n>=0 I have solved parts a and b and stuck on c (a) P is the...

Hello, In this problem I am trying to Determine the exact set of natural numbers n for which the inequality 2^n>=(n+1)^2 holds. (equation (1)) I have already dealt with the base case where n=6, (since the inequality does not hold for n<6), and so (1) holds for n=k, and I need to show that it...

Homework Statement \forall n \in Z^+, \sum_{i=1}^n \frac{\sqrt{i+1}}{2i} > \frac{\sqrt{n}}{2} Homework Equations I have to prove the above via mathematical induction. The Attempt at a Solution I did the base case, n = 1 and found it true for the base case. Then I assumed that...

-------------------------------------------------------------------------------- 1. Homework Statement Let H be a ten- element set of potive integers ranged from 1 to 99. Prove that H has two disjoint subsets A and B so that the sum of the elements of A is equal to the sum of the elements of B.

Homework Statement Let H be a ten- element set of potive integers ranged from 1 to 99. Prove that H has two disjoint subsets A and B so that the sum of the elements of A is equal to the sum of the elements of B.

Homework Statement The Fibonacci numbers are defined by f(1)=1, f(2)=1 and for n>2, by f(n) = f(n-2) + f(n-1). Prove by mathematical induction that f(3n) is even for all natural numbers n. Proof: Base case: (n=1) f(3) = f(2)+f(1) =1+1 =2 is even Induction hypothesis: (suppose the...

This is not homework, per se, but I have recently started reading Courant and Robbins' What is Mathematics? Homework Statement Prove by mathematical induction. (1+q)(1+q2)(1+q4) ... (1+q2n) = (1-q2n+1) / (1-q) Homework Equations Only the problem itself.The Attempt at a Solution Using the...

Homework Statement Prove by matematical induction that (2^(n+1)+9(13^n)) divides by by 11 for all positive intergers Homework Equations The Attempt at a Solution I really have no idea where to start...

Homework Statement Is mathematical induction a deductive or inductive argument? Would appreciate the help. Thanks. Jeremy Homework Equations The Attempt at a Solution Its name suggests that the process is inductive, yet I know all of mathematics depends on deductive reasoning...

Hi, I need some help with mathematical induction The question is as follows: prove that \sum_{i=0}^n (i-3) \geq \frac{n^2}{4} I have shown that it holds for the base step where n=12 \frac{144}{4} = 36 and the sum of all the i's up to 12 -3,-2,-1,0,+1,+2,+3,+4,+5,+6,+7,+8,+9 = 39...