https://youtu.be/Y11HGQ1LltI
The part where the general terms of the AP are defined. I don’t understand how the formula for sum of the natural numbers was arrived at.
Thanks.
In: Mathematics
For clarity (I didn’t watch the whole video), are you meaning the formula Sn=(n/2)x[2a1 + (n-1)d]?
If so, it goes like this.
For an arithmetic series, there is an initial seed number a1 (which I’m just going to call a for now) and a difference term d. Each term is made by adding d to the previous term. So the nth term is given by an=a+(n-1)d.
We okay with that?
If so, one thing that we can notice is that we can pair up the numbers. Let’s take the sequence of natural numbers 1,2,3,4,5,6,7,8,9,10,…
Suppose we want to add up the first n terms. We could just do that the long way. Alternatively, we could be clever. If I pair up the numbers starting from both ends, I get that 1+10=11, 2+9=11, 3+8=11, 4+7=11, 5+6=11. So, I now have a bunch of pairs that equal 11.
One way to see this is to take the nth term a+(n-1)d and add it to the first term a. We get 2a+(n-1)d. The next pair would be a2=a+d and a(n-1)=a+(n-2)d. Adding them, we get 2a+(n-1)d, again. Now, how many pairs? Well, we had n numbers which we paired, so we’re going to have n/2 pairs.
So we have n/2 pairs, each of which adds to 2a+(n-1)d, so the total is (n/2)x[2a+(n-1)d]. For the example above, S10 would be 5×11=55.
There’s a famous (probably apocryphal) story about the mathematician Gauss, when he was at school. His teacher was getting annoyed because he was too quick to finish the work. So the teacher said “Fine, if you’re so smart, add up the numbers from 1 to 100.” The teacher looked over a minute later to see Gauss doing nothing. They enquired, and Gauss said “I finished. It’s 5050.” Gauss had used this method, and saw that 1+100=101, 2+99=101,… and 50×101=5050.
For future reference, you might find r/math helpful. There’s a recurring post where people can ask questions 🙂
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