Proof by Induction : Sum of series ∑r² | ExamSolutions

Here you are shown how to prove by mathematical induction the sum of the series for r squared. ∑r²

[Sarank]

10 years later and this goat is still carrying further maths

[mrboyban]

If you hate proof by induction put your thumbs up

[thomaslee]

You sir are a genius! This video is flawless and helped me grasp the concept in one go! Thanks a lot!

[dhvsheabdh]

I assume you're wondering how the hell he's changed the "(k+1)^2" into 6(k+1)?
Well: Already, from the equation we can see that there are essentialy two things being added together. We have the Sum from r=1 to k of r^2, equal to k/6(k+1)(2k+1) and (k+1)^2. The thing is, we want to add both of these things toegther. If I make it more simple, it's like trying to add a/6 and b in one bracket without any fractions. It is neccessary to know that b is equal to 6b/6. So (see next comment...)

[cooldude]

Thank you, you did a much better explanation than my teacher who thought it would be a good idea to rush through the last chapter of FP1

[iamkaja]

OH MY GOD THANK YOU. Exam tomorrow - why didn't I discover this earlier!? Better late than never though, thank you so much!

[jusyungb]

In a paper I first found out wat I was working to towards then realised I couldn’t factorise it towards it, not sure why. But then I worked from rhs to lhs such as expanding the rhs to show it equals to my lhs. Will I still get the Marks cus it’s proved ?

[nuse]

I do not see where you get 1/6th from both sides the addition sign.

[julesofharbledown]

Immensely helpful, thank you. I was missing a step the textbook just didn't explain well.

[lavarball]

great video. but this induction stuff is longggg

[dhvsheabdh]

So...(k+1)^2 is equal to 6 lots of this, divided by 6. But BECAUSE it has been divided by 6, just like the (1/6)n(n+1)(2n+1), they can now be added together for simplification. However, another common factor in both of these terms to be added is (k+1), so he's decided to take that out also. So if you take out a sixth and (k+1) out of both of the things to be added, you end up with (1/6)(k+1) as the thing outside of the brackets. However, you're probably also wondering where the lone k has gone..

[BillStrait]

I'm taking a discrete math course online. This exact problem is on this week's assignment. For the first time since the course began, I feel I understand what's happening. Thanks a lot.

[GabbersLJ]

Oh my word that was so helpful thank man I was lost as he'll and you have shown me the light, thx so much.

[cary_domiii]

I've been struggling with this problem to find our that my only mistake was that I forgot to square (k+1)....

[AaaAaa-ybnb]

A huge help. Thanks for the clear explanation. FINALLY understand each step.

[BorisMediaProds]

You kind of sound like Raoul Silva from skyfall.

[dhvsheabdh]

well, the lone k dissappearance is simple. In the equation you can see that (1/6)k(k+1)(2k+1). They are all to be multiplied together, so it makes no difference in which order they are. So abcd = dabc = dcba etc. So he's moved the k infront of the (2k+1), so it makes it seem more simple as he multiplies everything by (1/6)(k+1). So, as (1/6)(k+1) is a factor of (1/6)(k+1)k(2k+1) and (1/6)[6(k+1)^2)], they can be added together, and hence simplified. Message me if you have any problems :)

[nini]

Thank you
Now i really understand the topic

[Soojene]

The exam solutions guy is an absolute lad!

[rosin]

Great explanation! This was really easy to follow and understand.