If you have any math discovery or fun problem that you want me to present in a video, please leave it here (with solutions hopefully). I will try my best to make it into a video for you and will credit you in the video.

I wanted to see if I can put a hypothetical situation into an equation. I was thinking it dealt w weighted averages & possibly system of equations. But I wasnt sure about the system of equations part. I think it deals with weighted averages b/c of the consumer price index/basket of goods was something I used to put together a financial map b/f. So this was possibly going to be a continuation of that financial map.

I've also started my research w/ the consumer behavior/economics part. I got an idea from acctng bc of journal entries & bal sheets that making an equation for certain things would be a really cool thing & useful when considering what things to buy while on a pre-existing budget, how much to buy, & maybe when to buy those items. Also I wanted to be able to view in equation form how a budget would be affected. I have a spreadsheet already created for food bill. Which is 1 part of the equation/s I think

Dear Steve, the internet is full of videos on the topic of euler's number, e. However I find that there is something that I don't like about most explanation videos: instead of explaining e, most videos end up explaining exponential functions in general. That's why I would like to present the following explanation, which I think is far more clear:

First consider trying to find the derivative of

f(x) = 10^x

We do this from scratch, starting with the definition of a derivative

f'(x) = 10^(x+h) - 10^(x) / h

= 10^x * (10^h - 1)/h

Note here that the first term 10^x is simply f(x), and that the second term (10^h-1)/h is simply f'(0). Here we are stuck, because we don't have any method to tell what the value of f'(0) is.

WOULDN'T IT BE NICE if f'(0) would just nicely equal to 1? That would solve all our problems for sure! Instead of 10^x, consider a more general form a^x. We can safely claim that there must be some value of "a" where f'(0)=1. To see why this must be true, have a look at https://www.desmos.com/calculator/ukstxtoen1

If you want, you can then go on about methods to find a good approximation for e, since it is just a number on the number line. But I'm just going to leave it here.

Summarizing, we have "invented" Euler's number because this was our only hope at every finding the derivative of an exponential function. No need to convince anyone why it is useful.

I hope you like this approach, let me know what you think if you see this :)

A1 buys grocery worth rs x1 from A2 and gives rs y note(y>x1) , A2 goes to A3 and buys grocery worth rs x2 and gives A3 note of y, A3 goes to A4 and buys grocery of worth Rs x3 and gives same note to A4, A4 does the same process with A5, A5 with A6,..., An-1 with An. At the end of this chain An gives change of y-xn-1 to An-1, An-1 gives change of y-xn-2,... A2 gives change of y-x1 to A1. A1 goes away taking the grocery and all Ai taking their groceries.Now at the end of this cycle, An notices that note of worth rs y is duplicate and goes to An-1(note- Ai is shopkeeper for i>1 and A1 is primary customer) to take rs y, An-1 goes to An-2 to take y,....A3 goes to A2 to take y.Now what are the losses of Ai?

Note :n>=3.

Solution Hint:Net profit/loss in such a system of money transaction is zero. Take n=3, profit of A1=y, Profit of A3=0, So profit of A2+profit of A1+profit of A3=0, implies profit of A2=-y.

Hi! There is this cool sequence I found recently: 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ... . My challenge is: Can you find at least three different formulas for the n-th term—one formula based on the triangular numbers, one formula using an integral, and one formula being recursive?

Thanks for your great videos, which are cheering me up every day :)

Hi blackpenredpen, I was recently watching your videos on 1+2+3+4+5+...=? , and I was trying to figure something out with that. But instead of doing that, I just chose the consecutive odd numbers. I can’t type math on here, so I typed it in a google doc and I will put a link here if anyone wants to check it out. Also if you like it blackpenredpen, could you make a video about it? Thanks, The link for the google doc is here: https://docs.google.com/document/d/1FNeiXOtaue_yJSJZUzV-F2I7G7gqoqgiNz6qCr2ewQs

I was thinking: what would happen if f(x)=sqrt(x) and then f(x) is substituted for the Fourier Series and Parseval's theorem? Could we finally get an expression of the Riemann zeta function of 3 and possibly other odd positive integers greater than 1? While I have done some work, it uses 2 new integrals: the Fresnel sine integral and the Fresnel cosine integral. Thus, it's highly recommended that you, Mr. Cao, make a video about these integrals and then make a video to see what will happen if f(x)=sqrt(x). Thank you! :D

I made 2 videos about summing over the 0'th to n'th derivatives of a function and it gives some really neat results including a elegant proof for the integral form of the gamma function for positive integers. If you're interested in sharing my findings, feel free to do so! I hope you'll enjoy them even though they are low quality videos. This is the url for my youtube channel: https://www.youtube.com/channel/UCkE6Jod-AmyDxU78eUz2omg

Could u connect algebraic problems to geometry? I've done my coursework about connection between trigonometry and geometry and also about trigonometric series of sum. If it's interesting for you, I can send u some excellent moments from my work.

The specific identity that I'm looking forward to you to prove is this:

Founded by American Mathematician Andrew M. Gleason, he used it to construct the regular heptagon.

https://en.wikipedia.org/wiki/File:01-Siebeneck-Tomahawk-Animation.gifHey, do u create custom exercises that deal with consumer behavior?

It's only enough for a tiktok but I found this and left it in a youtube comment

Explanation:

2 = 1+1

2 = 1+(3/3)

2 = 1+(3/2+1)

2 = 1+(3/2+(4/4))

2 = 1+(3/2+(4/3+1))

2 = 1+(3/2+(4/3+(5/5)))

2 = 1+(3/2+(4/3+(5/4+(6/...))))

Dear Steve, the internet is full of videos on the topic of euler's number, e. However I find that there is something that I don't like about most explanation videos: instead of explaining e, most videos end up explaining exponential functions in general. That's why I would like to present the following explanation, which I think is far more clear:

First consider trying to find the derivative of

f(x) = 10^x

We do this from scratch, starting with the definition of a derivative

f'(x) = 10^(x+h) - 10^(x) / h

= 10^x * (10^h - 1)/h

Note here that the first term 10^x is simply f(x), and that the second term (10^h-1)/h is simply f'(0). Here we are stuck, because we don't have any method to tell what the value of f'(0) is.

WOULDN'T IT BE NICE if f'(0) would just nicely equal to 1? That would solve all our problems for sure! Instead of 10^x, consider a more general form a^x. We can safely claim that there must be some value of "a" where f'(0)=1. To see why this must be true, have a look at

https://www.desmos.com/calculator/ukstxtoen1If you want, you can then go on about methods to find a good approximation for e, since it is just a number on the number line. But I'm just going to leave it here.

Summarizing, we have "invented" Euler's number because this was our only hope at every finding the derivative of an exponential function. No need to convince anyone why it is useful.

I hope you like this approach, let me know what you think if you see this :)

Kasper Arfman

Sir I am stuck on this problem for months can you please help me in solving it. It would be a pleasure.

Dear Omair siddique, here is a solution to your problem:

Maybe a tribute to John Conway who died recently of Covid-19

Pell's equations through an example ?

I like this subject but it is difficult.

A1 buys grocery worth rs x1 from A2 and gives rs y note(y>x1) , A2 goes to A3 and buys grocery worth rs x2 and gives A3 note of y, A3 goes to A4 and buys grocery of worth Rs x3 and gives same note to A4, A4 does the same process with A5, A5 with A6,..., An-1 with An. At the end of this chain An gives change of y-xn-1 to An-1, An-1 gives change of y-xn-2,... A2 gives change of y-x1 to A1. A1 goes away taking the grocery and all Ai taking their groceries.Now at the end of this cycle, An notices that note of worth rs y is duplicate and goes to An-1(note- Ai is shopkeeper for i>1 and A1 is primary customer) to take rs y, An-1 goes to An-2 to take y,....A3 goes to A2 to take y.Now what are the losses of Ai?

Note :n>=3.

Solution Hint:Net profit/loss in such a system of money transaction is zero. Take n=3, profit of A1=y, Profit of A3=0, So profit of A2+profit of A1+profit of A3=0, implies profit of A2=-y.

Can you decide this problem

U should do 100 differential equations.

Hi! There is this cool sequence I found recently: 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ... . My challenge is: Can you find at least three different formulas for the n-th term—one formula based on the triangular numbers, one formula using an integral, and one formula being recursive?

Thanks for your great videos, which are cheering me up every day :)

Hi blackpenredpen, I was recently watching your videos on 1+2+3+4+5+...=? , and I was trying to figure something out with that. But instead of doing that, I just chose the consecutive odd numbers. I can’t type math on here, so I typed it in a google doc and I will put a link here if anyone wants to check it out. Also if you like it blackpenredpen, could you make a video about it? Thanks, The link for the google doc is here:

https://docs.google.com/document/d/1FNeiXOtaue_yJSJZUzV-F2I7G7gqoqgiNz6qCr2ewQsVideo on calculating the area of a shadow from an object.

I was thinking: what would happen if f(x)=sqrt(x) and then f(x) is substituted for the Fourier Series and Parseval's theorem? Could we finally get an expression of the Riemann zeta function of 3 and possibly other odd positive integers greater than 1? While I have done some work, it uses 2 new integrals: the Fresnel sine integral and the Fresnel cosine integral. Thus, it's highly recommended that you, Mr. Cao, make a video about these integrals and then make a video to see what will happen if f(x)=sqrt(x). Thank you! :D

Solutions to 100x=2^x

I made 2 videos about summing over the 0'th to n'th derivatives of a function and it gives some really neat results including a elegant proof for the integral form of the gamma function for positive integers. If you're interested in sharing my findings, feel free to do so! I hope you'll enjoy them even though they are low quality videos. This is the url for my youtube channel:

https://www.youtube.com/channel/UCkE6Jod-AmyDxU78eUz2omgif you want to contact me, you can email me at

postvoortygo@hotmail.com:)Could u connect algebraic problems to geometry? I've done my coursework about connection between trigonometry and geometry and also about trigonometric series of sum. If it's interesting for you, I can send u some excellent moments from my work.

Respected sir,

I would like to know the below.

if you differentiate any constant other than 0, i.e. n(x^0) [n€N], we get 0.

But what will we get if we differentiate 0, i.e. x^(-infinity)????