Need help understanding the last part of the induction regarding (n(n+1)(2n+1))/6

Question

I am looking at the induction proof of $$ \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} $$

So i do the basis case which holds up.

Then i move on to the induction step and uses k+1 instead of n and starts off like so:

$$ \sum_{i=1}^{k+1} i^2 = \frac{k(k+1)(2k+1)}{6} + (k+1)^2 $$

Then i take out a common factor:

$$ (k+1) (\frac{k(2k+1)}{6} + (k+1)) $$

Then i put it all on the same fraction(is that the right term ?) and get:

$$ (k+1) (\frac{k(2k+1)}{6} + \frac{6(k+1)}{6}) $$

and get

$$ (k+1) + \frac{2k^2+7k+6}{6} $$

My question now is, how does my textbook go from this to the following:

$$ \frac{(k+1)(k+2)(2k+3)}{6} $$

I was thinking about just doing something like, replacing k with 2 then i would get :

$$ \frac{2*2+7*2+6}{6} + \frac{2*2+7*2+6}{6} + \frac{2*2+7*2+6}{6} $$

.

I have a few questions, and i am not that strong in even basic math e.g. i had to look up how to multiply a number with a fraction, so i would really like a answer like i am 5.

1. Is there any general term, e.g. like i used the "take out the common factor" that i can use here ??
2. At first i thouth i would arrive at a solution that looked like $$ \sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} $$ And it kind of does, but if i had to do this by my self i wouldn't be able to know that i had arrived at the end result, what is it exactly that the result should look like? i was thinking about that the solution should basically just be the initial statement but with n+1 instead of n, can someone explain that in a more formal manner? because its hard to solve this if you are not even sure what you are looking for.
3. A more general question, that is probably way over this scope, there is a lot of factorizing in this, is there any good resources that looks at a lot of induction proofs like the person doesn't have a solid background in math? or would it help to just go back into college/high school math and read up on the fundamentals of equations?

Answer 1

See that:

$$\begin{align}(k+1)\left( \frac{k(2k+1)}{6} + (k+1)\right)&=(k+1)\left( \frac{k(2k+1)}{6} + \frac{6(k+1)}6\right)\\&=(k+1)\left( \frac{2k^2+k}{6} + \frac{6k+6}6\right)\\&=(k+1)\left( \frac{2k^2+k+6k+6}{6}\right)\\&=(k+1)\left( \frac{2k^2+7k+6}{6}\right)\\&=(k+1)\left( \frac{(k+2)(2k+3)}{6}\right)\\&=\frac{(k+1)(k+2)(2k+3)}6\end{align}$$

If your having difficulty with the factoring step, I would review how to factor quadratics.

Answer 2

You forgot a pair of parentheses on factoring $k+1$. You should have written $$(k+1)\biggl( \frac{k(2k+1)}{6} + (k+1)\biggr).$$

Answer 3

Not entirely sure on what question 1 is about, but I believe the answer is $n=1$. This is the simplest and the start of the induction.

For question 2, see that if we know our formula works for $n=1$, we have proven that it works for $n+1$.

In this case, $n+1=2$, so our formula works for $n=2$ now. But we also proved it works for $n+1$, which then becomes a proof for $n=3$. etc. we have proved it for $n=1,2,3,4,\dots$

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Induction is not a process that solves the problem, rather it shows that some solution is indeed the solution. The general steps are to show it is true for $n=1$, then for all $n\implies n+1$, the proof follows true for $n=1,2,3,\dots$