The best way to write the number ____.

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[איילבןדוד-דפ]

I solved a bit differently (but same idea)

1. Multiply the sum by 2
2. Replace every 2 in nominator by (3-1), (5-3), …, (121-119)
4. Then divide denominator to get same as in (2) above but with square root.
5. Telescopic sum collapses and we have only 11 – 1 left which is twice the sum.

Less general for sure but without the need to re-index sums.

[MDMajor]

Interestingly, if you set an upper bound of 2k(k+1) for some natural number k, this sum is always equal to k. 2k(k+1) also happens to be 4 times the kth triangle number!

[goodplacetostop]

6:26 Solving the problem without having to clean the board 😎

[beatrizkarwai]

could you give some lectures on galois theory? that'd be pretty cool

[itays]

It's probably gonna be about a "conjugate" and a telescoping sum, I'd have an educated guess that the answer is 1/2*(sqrt(121)-sqrt(1)) = 5.
Edit: yup.

[Drk]

Nice! I thought the same approach in the bus and checked it at home. Interesting how a sum of irrational numbers turns into an integer ❤️

[SafetyBoater]

If the answer is 5, this is the fastest mental solve of one of your questions I've seen yet.
After watching: after rationalizing the first term the telescoping cancelations are obvious. Every term has -2 in the denominator and the numerator collapses to the first minus the last.

[criskity]

This is a great illustration of how the sigma summation notation makes things easier.

[LightPhoenix]

I did essentially the same thing without invoking sums formally. Multiplying each term to simplify the denominator have -2, and the numerator terms would all cancel to 1-11.

[kujmous]

So if we wanted the answer to be 6, we'd have to extend the sum up to √169. That means the difference in those sums would be 1, and that would hold true for as far as we wanted to go.

[AndyGoth]

At first glance I thought, "oh a telescoping series!", but then was disappointed to find that everything was positive. Yet in the end, you were able to make it behave as a telescoping series anyway! Magnificent!

[manucitomx]

This was fun.
Thank you, professor.

[MM-wjpz]

You can easily extend this to arithmetic progressions!

[sharpnova]

this is a nice alternative way of representing a telescoping series

[michaelgolub]

Really beautiful and expressive though the idea of rationalizing was quite obvious (I thought of it at the beginning, for common denominator seemed useless).

[ZiyadAllawi]

It’s 5, mentally in less than 30 seconds!
(1-11)/-2=5.

[lisandro]

You did one similar before, the end points were sqr of 2020 and sqr of 2021

[MrRyanroberson]

this is the earliest i've been, and it's for a really simple one

[RAG]

You managed to make it look a lot more complicated than it was. It is a pretty straightforward telescoping sum. You hid this with the sigma notation.

[trueriver]

By around 1:19 when you used the word "overlap" I was already thinking "the trick is going to be telescope".