Arithmetic series and abstract formulas
You might have heard the story about Gauss who, in 5th grade or so, was given an annoying task by their teacher to sum all the numbers from 1 to 100. While the rest of the class was diligently writing out their huge sum, Gauss leaned back with their arms behind their head, having calculated the answer 5050 within mere minutes. Gauss got rewarded with a beating by the teacher who accused Gauss of cheating.
If you haven’t tried to deduce this on your own, try now before you read further. That is, think of a shortcut for summing the numbers 1 to 100.
Here’s the quick formula for summing all the numbers to n to beat the teacher:
The pattern of 1 + 100, 2 + 99, 3 + 98 all adding to 101…it’s genius! I think of it as n/2 pairs of terms that add to n+1 so I would never forget the formula.
But what if n is an odd number, like 101? Then we don’t have n/2 pairs. Or do we? For 101, we have 50.5 pairs, which makes sense, as we have 50 pairs that add to 102, and then a “half pair”–the left over number in the middle, 51. Alternatively, we could use the formula to calculate the sum of n-1, which is an even number, and then just add n to it.
The algebra works out. The intuitive understanding still works, but we had to stretch it. Let’s stretch it even more. What if we want to sum this arithmetic sequence:
-5, -4, -3, -2, -1, 0, 1, 2, 3, …, 100
Or this one:
1, 1.5, 2, 2.5, 3, 3.5, …, 100
In both cases the formula won’t work. It only works for a sequence of natural numbers starting from 1. Here’s another formula that works for all arithmetic series[2]:
a{0} stands for the first term in the series, a{n} for the last. n is the number of numbers in the sequence to sum. The initial formula for the arithmetic series of natural numbers starting with 1 is now just a special case where a{n} is n and a{0} is 1.
How did they get this more abstract formula? Take some time to think about what this formula is saying before reading on.
The thing that characterises an arithmetic sequence, is that it increments always by the same amount. The sequence of numbers is like a triangle [1].
The series, which is the sum of the sequence, is then like the area of this triangle. The area of this triangle is the rectangle made by the average number and the base. The average number is the middle number, or the number equidistant from the first and last numbers, because the sequence increments uniformly. So now the formula makes sense:
Given that the average of a bunch of numbers is the sum of the numbers divided by the number of numbers, the formula for an arithmetic series is just an algebraic manipulation of that fact.
But wait…isn’t the formula of a triangle b*h/2? So shouldn’t the formula be a{n}*n/2? When a{0} is zero, this is the case. But otherwise, the area is more like this:
The area is the sum of triangle with height a{n}–a{0} and the rectangle with height a{0}. If you do the algebra, it works out that the area is our formula:
Which is quite interesting, because it’s saying that the area of the trapezium shape thing above is equal to the area of the triangle whose height is the sum of the two sides. So many interesting insights. If you have any others, please tell me!
[1] Thanks Chris Berkhout
[2] Thanks Khan Academy
Thanks http://mathurl.com/ for the formula images.
Why am I thinking about this? Because I’m thinking about geometric series and I first wanted to understand arithmetic series.
