MHB Solving Math Induction: Step-by-Step Guide

[delc1]

Could anyone please help with this question regarding mathematical induction;

View attachment 2667

step by step procedure/ solution would be greatly appreciated. Thanks!

 

[chisigma]

Could anyone please help with this question regarding mathematical induction;

View attachment 2667

step by step procedure/ solution would be greatly appreciated. Thanks!

Let suppose that...

$\displaystyle 3| \{n\ (n^{2} + 5)\} \implies n\ (n^{2} + 5) = 3\ k\ (1)$

In thi case is...

$\displaystyle (n+1)\ (n^{2} + 2\ n + 6) = n^{3} + 5\ n + 3\ n^{2} + 3\ n + 6 = 3\ k + 3\ n^{2} + 3\ n + 6 \ (2)$

... and clearly 3 devides the (2). For n=1 $\displaystyle n\ (n^{2} + 5) = 6$ so that...

Kind regards

$\chi$ $\sigma$

 

[delc1]

Still not quite sure what you mean?

I know how to do the base step, but the induction step is challenging

 

[Evgeny.Makarov]

I think your difficulty is not with this problem in particular, but with applying induction in general. Therefore, ideally you should get a description of proof by induction and ask questions about that, possibly using the example you gave. For example, "I am not sure how to state the induction hypothesis in this problem. Is the following correct?"

For an outline of induction proof, see https://driven2services.com/staging/mh/index.php?posts/45490/.

 

[MarkFL]

Still not quite sure what you mean?

I know how to do the base step, but the induction step is challenging

For the inductive step, I would let:

$$f(n)=n\left(n^2+5\right)$$

And then consider adding:

$$f(n+1)-f(n)$$

to both sides. Add this as is to the left side of your hypothesis, and add the simplified form to the right, and you should find your proof is complete.

 

[Deveno]

I'd like to make it clear to you what you have to do, in order to prove this by induction.

First, we ASSUME it is true for $n$, that is we assume:

$n^2(n+5) = 3k$, for some integer $k$ (we don't have to really know which one).

Let's re-write this in a form that will be more easy to "spot" later on:

$n^3 + 5n = 3k$.

We need to USE this somehow, to prove that under these circumstances for $n+1$:

$(n+1)^2((n+1) + 5) = 3k'$ (again, all we need to do is show $k'$ exists, we do not need to find it specifically).

Let's re-write this in a form that will be more helpful to us:

$(n+1)^3 + 5(n+1) = 3k'$ <---this is what we want to prove.

I suggest multiplying out the left-hand side, and see if you "recognize" some part of it (like maybe the part we use in our assumption (induction hypothesis)). What can you say about what's "left over"?