An arithmetic *sequence* is a sequence of numbers with every consecutive pair having the same difference. For example,
```
1, 4, 7, 10
```
is an arithmetic sequence because `4 - 1 = 3`, `7 - 4 = 3` and `10 - 7 = 3`. An arithmetic *series* is the sum of an arithmetic sequence, for example
```
1 + 4 + 7 + 10.
```
The sum of an infinite arithmetic series is not a number, so the question 'what is the value of an arithmetic series' is only interesting for finite arithmetc sequences/series, so we only focus on these here.
The positive integers up to (and including) 100 is another arithmetic sequence, and we can ask what the corresponding series is, i.e., what is
```
1 + 2 + 3 + ... + 98 + 99 + 100?
```
Famously, [Carl Friedrich Gauss](https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss) solved this problem at the age of 7 faster than anyone else in his class by noting a pattern,
```
1 + 2 + 3 + ... + 98 + 99 + 100
100 + 99 + 98 + ... + 3 + 2 + 1
----------------------------------------
101 + 101 + 101 + ... + 101 + 101 + 101
```
To find the sum we can take twice the sum instead and notice that every opposite pair `(1, 100), (2, 99), etc...` has the same sum `101`, with `100` terms in the series, so adding two coipes of the series gives `100*101`, then dividing by two to get the sum of the original series, the value is
```
100*101/2.
```
This idea immediately generalizes to showing
```
1 + 2 + 3 + ... + n = n*(n + 1)/2
```
for any positive integer `n`, as there are `n` terms pairing up with sums `n + 1`.
But our first example above did not start at 1, nor increment by 1, so what can we do here? If we simply shift the sequence to start at `a` instead of 1 then we have
However, in case one feels like this is just a trick and prefers algebraic manipulation, we can also prove this formula by [mathematical induction](https://en.wikipedia.org/wiki/Mathematical_induction).