Sum of an Arithmetic Progression (1 of 5: Visual derivation of formula)

[@thibautbelliere8057]

Most of the time, teachers just tell us to study the formula without explaining but you explain it so nicely wich it helps us to memorize it at the first view. Than you for your work!

[@saumyojitdas4212]

@Civic @Eddie Woo why at 2:04 sec he created duplicates of every terms or blocks. My thinking is that if he *did not created duplicates* ; till this; the statement will be this :1+3+5+7+9+11 instead of 2(1+3+5+7+9+11) 4:02-->then if i rearrange or mix the original blocks ( 1 block with 11 blocks) ( 3 blocks with 9) (5 with 7 ) there will be 3 columns instead of *6* Then he tells to intepret the "6 columns in his case / in my case it will be 3" as they are the same as the 1st column/(1st array of blocks of 1st term and last) . then equation will be 1+3+5+7+9+11=3(1+11) There is nothing to divide by 2 . It has come exactly perfect in both LHS AND RHS. Why he has created duplicates ? I know the intution is : he needs to derive the formula S = n/2 (first term + last term) so there is a "divide by 2" operation ...so to include this ; I have to count 1 extra of every terms. Then i can finally divide by 2 in the end just like he has done is there any other explantion?

[@Fas2Fun]

​@@saumyojitdas4212 you have 3 columns instead of 6. What you've done is you've folded the triangle onto itself. so you've derived the formula to be (no of elem in series) / 2 x (1+11) Since the triangle is folded in half, the number of colums/elements in the series gets halved. This when tested seems to work for other differences in the series and no of elements, same as the other formula did. So from your observation you arrive to the truth. 36 Suppose you do this for a very large numbers with very large series, the answer is incomprehensible to humans if they are not allowed to count manually or have no motivation to. You could think his actions as shifting or unbalancing and rebalancing the problem statement without changing the meaning of the problem. Simply put rephrasing the question to make it more humanly managable pieces. Eddie took a different path to rephrase the question is all. He creates duplicates so as to form rectangle with the series. For getting the total number of elements in a rectangle you know that multiplying sides gives the answer. So now you're dealing with just two numbers, which is easy to deal with. Then you divide by 2 because you've unbalanced the original value by double. i.e., to get back to the original value. You've done the same by halving the triangle. Same end is met via different means. (Breaking the problem into manageable pieces)

[@laisladelace]

The visual derivation pictures are sequentially logical, wonderfully color coded, and super clear. This video is still the best derivation of the sum for an arithmetic series video that I have been able to find ever since I renewed my studying of the derivation in 2018. Thank you so much for your work and for sharing it with the world!

[@50_shirsenduroy84]

hope I had a teacher like you, when I was young...

[@hafsaa4150]

I've never understood this formula but the visual has helped so much! Thank you :)

[@miguelfontenele221]

Other method teachers use to explain this formula is by explaining the way it was discovered by young Gauss (at least it was how the story was told to me) and for me it makes sense. Lets say you try to sum all integers from 1 to 100: 1+2+3+...+98+99+100 Gauss noticed that if you sum the first term with the last one (1+100) you get 101 and that's also true for the second term and the second from last term (2+99) which is also equal 101, that pattern keeps happening if the sequence is growing in a constant way (each number only increases by 1 in this case), so since you have 100 numbers in the series that means we gonna have 50 pairs so the sum is equal to 50 x 101 = 5050. 50 = total number of terms/2 101 = in this case first term summed with the last term Therefore we have: Sum = total number of terms/2 (first term + last term) You probably know about this already but just to throw it out there for anyone else that doesn't. Cheers! :)

[@hafsaa4150]

@@miguelfontenele221 Wow that's so interesting, thank you for sharing that. My teacher just told us to learn the formula but our class never understood where it derived from

[@miguelfontenele221]

@@hafsaa4150 Glad you found that interesting, learning the "why" of how things work is key for understanding any concept, it help a lot in fixating it to the mind in my opinion, always ask "why".

[@axn30158]

@@miguelfontenele221 What if the arithmetic progression has an odd amounts of elements? For example: (1,2,3,4,5) Using Gauss' idea: 1+5 2+4 What about the 3?

[@miguelfontenele221]

@@axn30158 Well, if the number of elements is odd then there is always gonna be a number right in the middle which is always gonna be "missing" if you use gauss method, but that number in the middle will always be the half of the first number + the last number of the sequence: (1+5)/2 = 3 1,2,3,4,5,6,7 ---> (1+7)/2 = 4 That always works with any arithmetic progression that has an odd number of elements, so if you're using gauss method you'd need to add the middle term everytime which can be calculated with that formula.

[@Abbe_Faaria11636]

I have been scratching my head on this topic for a while. I am a non-maths student, which made it more difficult. Thank you; this was a very insightful explanation. Hope to see more like this.

[@KSF_Foundation]

the best teacher. Eventhough English is my second language, ur explanation is good enough for me to understand.

[@prabhatkiranchaulagain1095]

Unfukin believable! Now That Is A Real Math Teacher Right There!!!

[@lacimoore2618]

This formula is also derived from the formula for a triangle. (1/2)bh or in this case (1/2)n(a+l). This is amazing!!

[@abhishekhmishra1111]

You are outstanding 👍 A lot of love from Nepal 🇳🇵 Most of the time teachers tell us to remember it, but now after this explanation I don't even gonna to remember it ... You make us understand beautifully

[@robert19]

actual legend. what a concise and understandable explanation and visualization. cheers!

[@leobinusting1202]

Great explanation!!! This visual explanation truly helps a student to understand arithmetic progression more deeply! Really great work! Salute u, Mr.Woo !

[@spandan_chavan]

Great teacher

[@shub9644]

If youre learning maths I highly recommend this channel!

[@whoisgliese]

This blew my mind, I didn't know there was such an intuitive way of understanding this formula

[@poey2811]

oh my god thanks so much this genuinely made it so clear to me

[@Someone-zx7xd]

The great thing is not to be able to understand it, but to be able to invent it. Don't be so happy if you understood it, be happy if you first discovered it and came here to find out that it is the same thing you discovered. Maths is not about memorization, but creativity. Understanding and remembering it is memorization but discovering it is creativity. I feel very bad that no matter how hard I think, I can never come up with these kinds of things.

[@jacksparrow-343]

Thank you so much teacher. This formula was draining my mental energy

[@scmtuk3662]

In the case of odd numbers: 1 + 3 + 5 + 7 + 9 + 11 + .... + (2n-3) + (2n-1) Duplicating this, and then adding the first and last term, we get: 1 + (2n-1) There are "n" columns, so this becomes n(1+(2n-1)) This then simplifies to n(2n) or 2n^2 Then dividing by 2 again gives us n^2. Hence the sum of the first "n" odd numbers = n^2

[@akkaunique3293]

Woww 🥺🥺❤❤ what a explanation 🖤 your the best teacher I had eer got.Wish I was a student of you.( Lv from 🇱🇰 🇱🇰 🇱🇰 Sri lanka)🥺

[@akkaunique3293]

🖤🖤🖤

[@gungeternal4119]

Dear Mr Woo, Might Seem Irrelevant to ask here but, I would really like a moment of your time if possible. First of all, thank-you for your diverse, effective, and passionate way of conveying the language and art form of mathematics to us, for an entire range of ages and cultures globally. It was powerful enough to change me from a fixed mindset of "I just can't do maths, English is all I have" to "I can maths, I love maths, give me the extension, I want to know more". But most importantly I am here because, I would be more than delighted to see a video from you about how natural intricacies and wonders, revealed by means of mathematics is testimony to Creation, as by means of further understanding mathematics and sciences, it becomes evident that there must have been a grand creator who architected the Natural Wonders. Thank-you SIr

[@rationalsceptic7634]

Nonsense...we invent patterns not God

[@sharmajika99percentagewall34]

0:50, a student admiring sir's well-maintained figure, NICE!!👍

[@roy-ub7sy]

Amazing visualization!

[@Jeffkingson]

The explanation is so clear. Thank you

[@RoniRonkoKovatch]

AWESOME!! :-)

[@davidramos4707]

This is a masterclass in participation

[@abhishekDhiman2000]

I wish you were my teacher when I was at school

[@youtubekings3853]

Saaaaame

[@harissiddiqui4306]

Bravo just amazing simply brilliant!

[@СагындыкАманжол]

Mr.Woo, very great visual represantation! May ask you what kind of software you use to make this type of presentations. Thanks

[@bebehyoda7676]

You have helped me soooo much but can you do a year 4 video because I am struggling on angles

[@awp7111]

Hi, I'm from Indonesia and I like your video 🤗🤗🤗

[@youtubekings3853]

I learnt this formula: Sum of n terms in ap: n/2[2a-(n-1)d]

[@ksrinivas1729]

Beautiful minds inspire others

[@RocketStudiosAdiCalv]

Big fan, much respect ✊

[@PratyooshBhatia]

🙌🙌

[@cowboyteacher5243]

Very nice sir. Keep it up

[@erikhai6225]

Keep up the good work sir! :)

[@CMAenergy]

Every video of yours someone is playing with the lighting and it affects what one sees on the board, The contrast is washed out with to much light,

[@wadimzeller8518]

My mind was blown

[@teja7986]

Thanks Woo

[@snowwolf4148]

Only if they’d taught math like this back in high school

[@bestkickers9245]

How u did the column chart animation?

[@raheelriaz786]

How can we get these slides which you have presented?

[@nelofarwali9982]

Can you please tell which software did you use??

[@sab_tech]

hi from India

[@kingnoob1372]

omg its eddie woo! I met him!

[@johnryder1713]

What software is he using there?

[@gaming_teddyyt9428]

Masha Allah, may Allah guide you to Islam!! 🎉

[@yasamienaraz3543]

Which yr is this for? I realised the people in the background are older than me. And what a weird way of putting it.. Duplicating then what not. I think I'd just add the answer together like the normal way... Or at leqst the expected way. That did not feel like six minutes. That hurts my brain.

[@anish7183]

eddie woo loves to talk and waffle. He's not concise or logical at all. I think he's a better teacher for students who are weaker at maths

[@boringtofu4433]

yeah I think some things shouldn't be outright told to the students, they should have to figure it out on their own otherwise critical thinking skills don't really develop, which can be problematic down the line.

[@subhash.n3680]

Vera leave

[@lfj270]

May I go to the bathroom please

[@ivankevinling1283]

:):):)