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
I suppose you know what APs are and how they work. In this case, we have an AP which are simply the natural numbers: “1,2,3,…”. We want to know two things from this progression:
1) What is the general term of the sequence? That is, what is the n-th number of the sequence?
2) What is the sum of the first n numbers in the sequence? That would be 1+2+3+…+n.
We can answer both questions for any arithmetic progression because there are know formulas for that.
The first one is easy, even if you don’t know the formula. Nonetheless, let’s do it with the formula: The general term “a_n” is
a_n = a_1 + (n-1)*d,
Where a_1 is the first term of the progression (in this case, it’s 1) and d is the diferrence of the progression (each consecutive term differs from the one before by one, because we are adding 1 each time, so d = 1). If we substitute the values in the formula we get
a_n = 1 + (n-1)*1 = 1+n-1 = n <=> a_n = n.
For the second one, there formula is the following: if we call the sum of the first n terms “S_n”, then
S_n = (n/2)*(2*a_1+(n-1)*d).
Again, simply substitute the values and you’ll get the formula for the sum of the natural numbers:
S_n = (n/2)*(2+n-1) = (n/2)*(n+1) (that is what appears in the video).
If you don’t care about using the formulas without knowing why they work, then you’re good to go. If you actually want to know how these formula for the sum was found, then, have a look at this: https://www.youtube.com/watch?v=NEWy3Ckv1Yw
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