This proving the proposition. What about k = 5? Suppose it is true for all $k\leq n$. $$. Free. However, sometimes strong induction makes the proof of the induction step easier. Proof: By induction. And thus easily show that it is divisible by 12. How can I label staffs with the parts' purpose. However, sometimes strong induction makes the proof of the induction step easier. Prove by induction that: I am trying to differentiate $$ f^{(k)}(x) $$ to get $$ f^{(k+1)}(x) $$ Hi ,I’m not a very bright math student who’s still trying to understand strong induction. Wouldn’t other numbers sort of work too? Just because a conjecture is true for many examples does not mean it will be for all cases. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Note that $(n+1)^4-(n+1)^2=(n+1)^4-(n+1)^2-(n^4-n^2)+(n^4-n^2)$. I found that a method I was hoping to publish is already known. $$f^{(k+1)}(x)=2^{k/2}(e^x\sin(x+k\pi/4)+e^x\cos(x+k\pi/4))$$ n = 1: 12|(14 – 12) = 12|(1- 1) = 0 = 012 The metaphor of dominoes also gave rise to the chosen post photo, which was kindly shared under the creative common license by Malkav. Proof By Induction Steps Examples Study Com. A question for the deeply intrigued: investigate the factoring of n^k-n^(k-2). The weak induction method failed. Proof by Induction. rev 2020.11.24.38066, The best answers are voted up and rise to the top, Mathematics Stack Exchange works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, $$ Show that … 3 . Both methods have the same logical strength when we apply them, since in order to get to the k’th domino we need to topple 1,2,3,…k-1 anyway. I have shamelessly stolen this example from Hammack since I think it brilliantly shows when strong induction is better to use. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. \\=2^{-1/2}\left(\sin\left(x+\frac{n\pi}4\right)+\cos\left(x+\frac{n\pi}4\right)\right).$$, To utilize induction, note that $${d\over dx}ae^x\sin(x+b)=a\sqrt 2e^x\sin\left(x+b+{\pi\over 4}\right)$$. Show this by factoring as [latex]2k(k+1)(2k+1)[/latex]. I’m not a math major but I sometimes need to write proofs for computer science classes and I find it excruciating. Go through the first two of your three steps: Loading... Save for later. But lets first see what happens if we try to use weak induction on the following: Base case: we need to prove that 12|(14 – 12) = 12|(1- 1) = 0, which is divisible by 12 by definition. Mathematical Induction. Not to beat a dead horse, but … Before I read the comments, I tried figuring why 12 | n^4-n^2. So this example fails to show the power of strong induction. The good news is that this does affect your proof. I have proven it for $n = 0$ and then assumed it be true $n = k$. To switch to the metaphor of dominoes again – in the weak induction we need to know that the previous domino is toppled, then the next one will topple as well. - MathHub. $$e^x\sin x=e^x\Im e^{ix}=\Im e^{(1+i)x}.$$, $$(e^{(1+i)x})^{(n)}=(1+i)^ne^{(1+i)x}=\sqrt2^ne^{in\pi/4}e^{(1+i)x}=2^{n/2}e^xe^{i(x+n\pi/4)}.$$, $$\sin\left(x+\frac{(n+1)\pi}4\right)=\sin\left(x+\frac{n\pi}4+\frac{\pi}4\right)=\sin\left(x+\frac{n\pi}4\right)\cos\left(\frac{\pi}4\right)+\cos\left(x+\frac{n\pi}4\right)\sin\left(\frac{\pi}4\right) A proof by mathematical induction is a powerful method that is used to prove that a conjecture (theory, proposition, speculation, belief, statement, formula, etc...) is true for all cases. f(x) = e^x \sin(x) Created: Dec 4, 2011. Better examples are abundant in group theory, f.e. Both parts are done by cases (1,2 modulo 3, 1,2,3 modulo 4) and there are 6 non-trivial cases to deal with, arguably easier and clearer than the proof by strong induction you’ve given. $$ However, for most other values of l we would end up with a final expression where it is hard to prove that the expression is divisible by 12. This is useful. 1 ≥ To guide readers, please state whether your answer handles: PS: I have seen many induction related questions, and very often the problem lies with the OP's lack of a proper methodology (or style) in writing the proof whereas the answers focus on the particular case of the OP's question. Strong Induction Example 1. Returning to the question in order to procrastinate, :), there’s an even simpler, immediate way to show that 12|(n^4-n^2) – just write n^4-n^2 = n*(n-1)*n*(n+1) and we’re done! But you don’t reach a level where you need to understand strong induction unless you are either pretty bright, or good at faking . The reason for defining l as k-5 is the fact that ((l+6)^4 – (l+6)^2 gives us a polynomial where we can isolate 12 from all parts. Works as well. We need to show that 12|((l+6)4 – (l+6)2) is true. Regarding your second comment, most of the sources I have checked to read up strong induction includes the base case to prove a starting point. n = 4: 12|(44 – 42) = 12|(256- 16) = 240 = 20 * 12 Considering but I still don’t get it! f^{(n)}(x)=2^{\frac{n}{2}}e^x\sin(x+\frac{n\pi}{4}) Asking for help, clarification, or responding to other answers. […] learned about strong induction, and had a couple questions.