We should introduce a bit of terminology.
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What's an arithmetic progression?
An arithmetic progression is a sequence
with a common difference between the terms.
We should see an example.
This might begin 5, 12, 19, 26,
33, and then keep on going. The general
form for the nth term will be 5 plus 7n.
Why is that an arithmetic progression? Well the difference
between each term to the next is seven. Right?
5 plus 7 is 12 12 plus 7 is 19 19 plus 7 is 26 26 plus 7 is 33, and so on.
We can write down a general formula for an arithmetic progression.
Well, here we go.
In general,
a sub n equals some starting number a sub zero
plus a common difference times n. Well, here's another question.
Why are these things even called arithmetic progressions?
Each term Is the arithmetic mean
of its neighbors. For
reals, let's look back at that example. Yeah, in our
example, 12 is the arithmetic mean or the average of 5 and 19.
So 5 plus 19 is 24, and 24 divided by 2 is 12.
Same goes for 19, 19 is the arithmetic mean or the average of its neighbours.
I'm going to write that down, alright, 19 is the average of 12 and
26. What's 12 plus 26?
It's 38. And 38 divided
by 2 is indeed 19.
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