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Am sorry if i might seem a little noob, but my lecturer taught us to prove a few proof of functions by induction and even gave a few examples e.g
$1 + 3 + 5 + 7 + ... + 2n-1 = n^2$

But seriously he did'nt event tell us what proving a function by induction is. Please if someone would provide a brief and well understandable intro to proving functions by induction is. I'd appreciate! And to use as an example, is that equation there. Thanks in advance

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    Wecome to Mathematics Stack Exchange. You can learn it, see the answer of this guideline: show the first case of the identity holds (that is the easy fact $1=2\cdot 1-1$), after presume that the identity holds for the case $k$ and then you need to show the case $k+1$ (but it is only an easy fact about brackets, see it!) $$ \left( 1+3+\ldots+(2k-1)\right)+(2(k+1)-1)=k^2+2(k+1)-1=(k+1)^2.$$ – 2017-01-02

1 Answers 1

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Suppose you have... something. Like $1+3+5+\dots+(2n-1)$. You notice that

$$1=1^2\\1+3=2^2\\1+3+5=3^2$$

And you want to prove the general statement.

Then, induction goes like this:

Suppose it happens to be true for $k$. That is,

$$\underbrace{1+3+5+\dots}_k=k^2$$

Then, use this to prove it is true for $k+1$.

$$\begin{align}\underbrace{1+3+5+\dots}_{k+1}&=\underbrace{1+3+5+\dots}_k+(2k+1)\\&=k^2+2k+1\\&=(k+1)^2\end{align}$$

So, if your formula is true for $k=1$ (it is, we checked above), then it must be true for $k+1$, i.e. it is true for $k=2$.

If your formula is true for $k=2$, then it must also be true for $k+1$...

etc.

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    where's the $(2k +1)$ on the right hand side coming from please? – 2017-01-02
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    @void_shad See how the $\underbrace{1+3+5+\dots}_{k+1}$ changed? I took the last number out, and it happens to be $(2k+1)$. – 2017-01-02
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    Oh sure, hadn't seen that! Thanks mahn, looks pretty easy – 2017-01-02
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    Nice use of mathjax. It looks attractive. +1 – 2017-01-02
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    @SimpleArt I am curious why you answered a question you voted to close? – 2017-01-02
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    Well, ignore me you may. Do you not realize that in having voted to close after you answered, it is reasonable that users might wonder whether you voted to close a question you answered, in order to prevent other users from posting an answer. Alternatively, if you deemed a question close-worthy, and then answered it, how sincere are any of your votes? – 2017-01-02
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    @amWhy Oh no, I didn't mean to ignore you, I was just away at the time of your previous comment. And I voted to close after noticing there was a good duplicate. So to the last question, I think my votes are sincere. I usually close as duplicate if I have answered a similar question in the past and/or know of a good duplicate, but if I knew neither and thought an answer would be good, then why shouldn't I post an answer? If by the time I post the answer and it is brought to my attention there does exist a duplicate, then it is within all reason to close. – 2017-01-02
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    Whatever....just lettin' ya' know why people around here are "curious" when answerers vote to close the question they already answered. Best... – 2017-01-02
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    ^_^ Yes, I understand, I'm not trying to ruin the site. Best regards to you as well. – 2017-01-02