Arithmetic Sequences: Review
What is it?
An arithmetic sequence is a sequence in which all terms are on an equal distance to each other. e.g. the heights of a step of a stair, or the angles of 1, 2, 3... hours on a clock.
In other words, if you’re climbing stairs, the distance between the next stair and the previous stair is equal to the distance between the next-to-next and next stair, and so on. Therefore, next step - previous step = 2 steps after - next step = 3 steps after - 2 steps after... or, in general terms: \(a_{n+1} - a_{n}\) remains the same.
Properties!
This common \(a_{n+1} - a_n\) which is always the same is called \(d\). The starting point is \(a\) and the number of steps is \(n\):
\(a\): starting point
\(d\): distance between next and previous step
\(n\): number of steps
For example, for the sequence \[0,2,4,6,8\] The value of \(d\) is 2, the value of \(a\) is 0, and the value of \(n\) is 5.
Clearly, the following equation makes sense:
\(a_{n+1} = a_n + d\) i.e. next step = previous step + distance
You can do anything you want in arithmetic sequences with this equation, but it’s generally preferable to have one more at your arsenal:
Sum
What if you add all the heights of your stair, i.e. add the whole arithmetic sequence together? Let’s visualize:
First, you have starting point. (e.g. 2)
Then, you add starting point to starting point + distance, two starting point and one distance. (e.g. 2 + 3 = 2 + (2 + 1))
Then you add two starting points and one distance to one starting point and two distances, making three starting points and three distances... (2 + 3 + 4 = 2 + (2 + 1) + (2 + 2))
and so on, but at all steps, note you have as many starts as many times you added (i.e. \(na\)) and 1 + 2 + 3 ... starting points, for which we have the formula \(S = \frac{n(n+1)}{2}\) (explanation for another post!).
So we can just add them and get the sum \(S = na + \frac{n(n+1)}{2}\).
To visualize, you can think of a right triangle above a starting rectangle, added together, which is actually identical to calculating the area of the stairs (advanced challenge: try googling ‘area of triangle’ and see if there’s any similarities!)
Therefore, the sum of an arithmetic sequence is start times number plus 1 + 2 + 3... times step distance (use this to derive the formula on will!)
That’s all for arithmetic sequences.
