Mathematical induction Definition and 215 Threads

I know this sounds kind of like a basic question, but why does the method of mathematical indcution work to prove things like sequences and such? All the other proof methods I have learned have made sense to me, and I can prove using logical truth tables or axioms, but I don't really get how...

Homework Statement prove, using mathematical induction, that the next equation holds for all positive t. \sum_{k=0}^n \dbinom{k+t}{k} = \dbinom{t+n+1}{n} Homework Equations \dbinom{n}{k} = {{n!} \over {k!(n-k)!}The Attempt at a Solution checked that the base is correct (for t=0, and even for...

While learning mathematical induction,an idea occurred to me. Mathematical induction is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if anyone statement in the infinite sequence of statements is true, then so is the next one...

Homework Statement Conjecture a simple formula for n = 2,4,6,8 Homework Equations Kind of... Tn = (1-\frac{1}{n^2})(1-\frac{1}{(n-1)^2})... to n=1 The Attempt at a Solution The pattern is 3/4, 5/8, 7/12, 9/16 I know that the top is increasing by 2 and the bottom is increasing by 4... but I...

Homework Statement All I am trying to do is the basic mathematical induction routine: n=1, n=k, n=k+1 and how n=k proves n=k+1. The problem I am having is with the algebra. 4+16+64+...+4^{n}=\frac{4}{3}(4^{n}-1) 2. Homework Equations and Attempts For n=1 4=4 That's OK For n=k...

Homework Statement If a0=1, and a1=2, and an=(a(n-1))^2/an-2 for n>=2, prove by induction that an=2^n for n>=0 Homework Equations The Attempt at a Solution (B) a0=1=2^0=1 yes is true a1=2=2^1=2 yes is true (I) ak=(2k-1)^2/2k-2=2k Is it true that what I solved...

Homework Statement Use Mathematical induction to prove that the following equality hold for any natural number n. Show your work step by step. 1+5+9+ ... +(4n-3) = n(2n-1) Homework Equations The Attempt at a Solution

I have to solve: 1/2n <= (2n - 1)!/(2n!) I have no idea how to approach this problem.. Any hints? Thanks

Okay i need some help understanding what induction is..I know that for some open statement you must prove thatif the smallest element in the set is true... every element in that universe is true... I know that you use the basis step for the smallest element. and for the induction step you must...

The ’moments’ <t**n> of the distribution p(t) are defined as: <t**n> = integral from (0, infinity) p(t).t**n dt (1) where ** denotes to the power of Show (analytically) that <t**n> = n!τ**n (2) Hint: Use integration by parts to show that <t**n> = nτ<t**n-1> (3)...

Find a simplifying expression for the product (1-1/2^{2})(1-1/3^{2})...(1-1/n^{2}) and verify its validity for all integers n \geq 2 I know how to do the second part of the question, but no idea how to approach the simplifying of the expression. Any tips?

Homework Statement Prove the following identity by mathematical induction: \sum_{i=1}^n \frac{1}{(2i - 1)(2i + 1)} = \frac{n}{(2n + 1)} Homework Equations The Attempt at a Solution Let P(n) = \sum_{i=1}^n \frac{1}{(2(1) - 1)(2(1) + 1)} = \frac{1}{(2(1) + 1)} P(1) =...

Hi this is my first post so here goes... Basically I'm studying maths and in a section called proof and resoning they have introduced mathematical induction. I have tried to follow the examples but I still can't make head nor tail of it really. It makes absolutely no sense to me at all...

I'm just starting to get the hang of Mathematical induction and I was wondering if you guys could please check this proof just to make sure it is correct. Before I start, I will use (n k) to represent the binomial coefficient. ----------------- Prove 2^n = 1 + (n 1) + (n 2) + ... + (n...

Prove that S1, S2, S3 are true statements 1+3+5+...+(2n-1)=n^2 S1=1= (2(1)-1) = 1^2 True S2=1+3 = (2(3)-1) = 5 which cannot= to the sum of our first 2 integers, which will make it false! S3=1+3+5 = (2(5)-1) = 3^2 True The problem is with S2 the book gave me an answer of 4=4 which is...

[SOLVED] Library item: Mathematical Induction reported The following item was reported by tiny-tim : Click here. He has given the following reason:

help guys i am really stumped on this question. prove that if "n" is an integer , then n^2-n+2 is even

Homework Statement Prove that, for all integers n =>1 \frac{1}{1*2} + \frac{1}{2*3} + \frac{1}{3*4}... + \frac{1}{n+1} = 1-\frac{1}{n+1} Homework Equations I am a little stuck on this question. :| The Attempt at a Solution

Homework Statement Use mathematical induction to prove, for all integers n >= 1 n^3 - n is divisible by 3 Homework Equations Found equations with addition but no subtract involved. The Attempt at a Solution Another question suggests you expand the brackets, then insert...

Homework Statement 2^n>n , n\geq 1Homework Equations The Attempt at a Solution n=1 2^1>1 2>1 n=k 2^k>k n=k+1 2^k^+^1=2^k*2>2k=k+k>k+1 2^k^+^1>k+1 Ok, I don't understand the part k+k>k+1 If I get k=1 (from the k\geq 1) 1+1>1+1 2>2 which is not actually correct. Any help?

[SOLVED] Proof by mathematical induction Homework Statement Prove by mathematical induction that for all +ve integers n,10^{3n}+13^{n+1} is divisible by 7. Homework Equations The Attempt at a Solution Assume true for n=N. 10^{3N}+13^{N+1}=7A Multiply both sides by...

I am trying to understand induction and not having much luck. Here is my problem as I understand it Problem: n > or equal to 4, 2^n < n! Step 1) Prove it works for n=1 no, n=2 no, n=3 no, n=4 yes 16 < 24 step 2) assume it works for n=k 2^k < k! Step 3) prove it works for n= (k+1)...

Homework Statement trying to prove left-hand side = right-hand side this is where I'm stuck: 1 - [1 / (x+1)!] + [(x+1) / (x+2)!] = 1 - [1 / (x+2)!] Homework Equations The Attempt at a Solution i tried this but can't get anywhere get a common demoninator: 1 -...

Homework Statement n \epsilonN=n\geq0 6 divides (n^{3}+5n) Homework Equations (n^{3}+5n)=6q The Attempt at a Solution by expanding and simplifying and later on substituting 6q in (n+1)^{}3+5(n+1) ive arrived at 6q+3n^{}2 +3n+6 then.. I am stuck...pls help...lot of thanks!

Homework Statement Prove: 1^{4}+2^{4}+3^{4}+...+n^{4}=\frac{(n)(n+1)(2n+1)(3n^{2}+3n-1)}{30}Homework Equations Umm, I am not using any.The Attempt at a Solution So my first step: 1) Check for n=1 1^{4}=\frac{1(1+1)(2(1)+1)(3(1)^{2}+3(1)-1)}{30}=\frac{(2)(3)(5)}{30}=1 2)Now if n=k...

This problem is from Apostol's Caclulus book vol 1 page 35 or 36 #2 Show that 1 - 4 = -(1 + 2) 1 - 4 + 9 = 1 + 2 + 3 1 - 4 + 9 - 16 = -(1 + 2 + 3 + 4) is true by mathematical in duction I get to this step but have problem figuring out how to finish it off -1^(n+1) * n^2 = -1^(n+1)...

i'm on the last part of this question involving mathematical induction and i can't get the left side to equal the right saide. can anyone help me out? right side: [(k+1)+1]! - 1 left side: (k+1)! - 1 + (k+1) + (k+1)!

Homework Statement I have been workin on a mathematical induction question and have run into trouble with the simplification. Homework Equations The Attempt at a Solution I know that the solution i am trying to reach is (k+2)!-1 but i do not know where to go from the...

In mathematical Induction, sometimes two variables are given, such as: "All positive integers n and all real numbers x >= -1." My question is do you solve this normally and just keep x as itself or do you have to expand it like you do with n, making it k + 1 etc etc. Thanks in advanced

DISCRETE MATH: Prove a "simple" hypothesis involving sets. Use mathematical induction Homework Statement Prove that if A_1,\,A_2,\,\dots,\,A_n and B are sets, then...

Homework Statement 1. Prove that if n is an even positive integer, then n³-4n is always divisible by 48. 2. Prove taht the square of an odd integer is always of the form 8k+1, where k is an integer. 3. Observe that the last two digits of 7² are 49, the last two digits of 7³ are 43...

Here's my problem: 1 * 2 + 2 * 3 + 3 * 4 + . . . + n( n+ 1) = n(n + 1)(n + 2) / 3The Attempt at a Solution I know that there is a Basis step and Induction step. For the Basis step I have the following but don't know if I'm on the right track: Basis Step 1 * 2 = 1 * 2 * 3 / 3 , which is...

I'm connected from a phone so it would be pretty hard to write out MY equation for An...here's the equation in words: square root of 3 over 4 + square root of 3 over 12 * ((9/5 -9/5(4/9)^n)... using your general expression for A_n and the iterative (what does that mean?) relation between A_n...

I need help getting started on using Mathematical Induction with this problem... so what should i do first? a + (a+d)+(a+2d)+...+[a+(n-1)d] = (n/2)[2a+(n-1)d]

ok I am really confused now topic says it all.. I am given 4n-3 = n(2n-1) using mathemadical induction proof that is true. P(1) both equal 1 P(k) 4k-3 = k(2k-1) = k^2 - k P(k+1) 4(k+1)-3 =(k+1)(2(k+1)-1) if i simplify it all i get that 4k +1=2k^2...

Proposition: 1*2*3+2*3*4+3*4*5+...+n(n+1)(n+2) = [n(n+1)(n+2)(n+3)]/4 Step (1): If n=1 then LHS (left hand side) = 6, and RHS = 6 Thus, P1 is true. Step (2): If Pk is true then k(k+1)(k+2) = [k(k+1)(k+2)(k+3)]/4 Now, k(k+1)(k+2) + [k+1]([k+1]+1)([k+1]+2) = [k(k+1)(k+2)(k+3)]/4 +...

Proposition: 1*2*3+2*3*4+3*4*5+...+n(n+1)(n+2) = [n(n+1)(n+2)(n+3)]/4 Step (1): If n=1 then LHS (left hand side) = 6, and RHS = 6 Thus, P1 is true. Step (2): If Pk is true then k(k+1)(k+2) = [k(k+1)(k+2)(k+3)]/4 Now, k(k+1)(k+2) + [k+1]([k+1]+1)([k+1]+2) = [k(k+1)(k+2)(k+3)]/4 +...

[k(k+1)(k+2)(k+3) + 4(k+1)(k+2)(k+3)]/4 factor this out... What's the common factor? How did you get there? (ok i hope it doesn't require expanding the polynomials :p) Again, would it be easier if i substituted every (k+x) by a different variable, where (k+1) would equal to variable 'A'...

Hello everyone. THis is my first proof to strong mathematical induction so im' not sure if its correct or not it seems it though but then again I wrote it. Any suggestions/corrections would be great! THanks Here it is! http://suprfile.com/src/1/3j34eh1/lastscan.jpg

Directions: Evalute the sum, for n = 1, 2, 3, 4, and 5. Make a conjecture about a formula for this sume for general n, and prove your conjecture by mathematical induction. StatusX helped me get the first part, so I know that is right, about making the...

Hello everyone. I've been looking at examples and I can't seem to see what they are doing. For example, the book has: Suppose that d1, d2, d3 ... is a sequence defined as follows: d1 = 9/10, d2 = 10/11. dk = dk-1 * dk-2 for all inegers k >= 3. Prove that dn =< 1 for all...

Hello everyone I'm having problems on this last part of mathematical induction. I have to show that the two equations are equal to each other. The book shows a few examples which i will show below. They are writing the kst term separately from the first k terms. Heres my problem firstly...

Help with Proof and Mathematical Induction problem Here is my problem I need to solve: "Prove that the statement: \frac {1}{5} + \frac{1}{5^2} + \frac{1}{5^3} +... + \frac{1}{5^n} = \frac{1}{4}(1-\frac{1}{5^n}) is true for all positive integers n. Write your proof in the space below." I don’t...

Hey there evryone I need some help with this problem as I don't know which direction to go with it. Prove by mathematical induction that (13^n)-(6^n) is divisible by 7. The Base Step is obviously ok... Then assume (13^K)-(6^K) is true Then have to prove (13^(k+1))-(6^(k+1)) is...

Hi there folks, I have just a small problem with a specific induction problem. The problem itself is: "Prove n! > 4^n, for all n >= 9." So here's my work: 1) Show true for n = 9 LS 9! = 362880 RS 4^9 = 262144 .:. LS > RS 2) Assume true for n = k i.e. Assume that k! > 4^k 3) Prove true for...

Show that the sequence given by An = n/2^n is a null sequence.. Hint: We have proved by mathematical induction that2^n >or equal n^2, n> or equal 5... pls help...

I have a problem when trying to prove n! >= 2^(n-1). My work: Assuming n=k, k! >= 2^k-1 (induction hypothesis). To prove true for n=k+1, (k+1)! >= 2^(k+1)-1 = 2^k Now considering R.S., 2^k = (2^(k-1))(2)I get stuck here. I don't know how to continue onwards...

Hi, I am having real problems to do a mathematical induction on the following:confused: : 1) [SIZE="4"]A[SIZE="2"]n=((3^0,5)/2)(1+(0,6(1-((4/9)^n)) I also know that: 2) 1) [SIZE="4"]An=[SIZE="4"]A[SIZE="2"]n-1+((3((4/9)^n)(3^0,5))/4) please help me!

The following problem is given in my algebra book: For all positive integers n, n(n+1)(n+2) is divisible by 6. Prove using mathematical induction. First: k=1, 1(1+1)(1+2) = 1(2)(3) = 6, which is obviously divisible by 6 Next: k(k+1)(k+2) And then: (k+1)(k+2)(k+3) From this I...

Hi, all. I'm working on some proof by induction problems. While I understand the concept, this one threw me for a loop. Let x_1=\sqrt{2} and x_{n+1}=\sqrt{2+x_n} Show that x_n < x_{n+1} I'd greatly appreciate help with this. Thanks, bailey