Find a1 and d for the arithmetic sequence with the given terms.
a20 = 13, and a25 = 33
a25 – a20 = 20.
There are 5 steps from a20 to a25, so we divide by 5 to get
d = 4.
Then we proceed as before:
a1 + 19d = a20
a1 + 19(4) = 13
a1 = -63.
Once we know we have an arithmetic sequence, if we start at a1 and take (n – 1) steps of size d, we end up on term an.
We can write this idea in symbols as
a1 + (n – 1)d = an.
This formula has three unknowns: a1, n and d. Like a buy two, get one free deal on cups of key lime pie yogurt, any two pieces of information about an arithmetic sequence tells us the third and final piece of information. So, if we know a1 and d we can find any term an we like. We can also work backwards from d and some term an (which gives us n) to find a1.