Find *a*_{1} and *d* for the arithmetic sequence with the given terms.

*a*_{20} = 13, and *a*_{25} = 33

Answer

We have

*a*_{25} – *a*_{20} = 20.

There are 5 steps from *a*_{20} to *a*_{25}, so we divide by 5 to get

*d *= 4.

Then we proceed as before:

*a*_{1} + 19*d* = *a*_{20}

*a*_{1} + 19(4) = 13

*a*_{1} = -63.

Once we know we have an arithmetic sequence, if we start at *a*_{1} and take (*n* – 1) steps of size *d*, we end up on term *a*_{n}.

We can write this idea in symbols as

*a*_{1} + (*n* – 1)*d* = *a*_{n}.

This formula has three unknowns: *a*_{1}, *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 *a*_{1} and *d* we can find any term *a*_{n} we like. We can also work backwards from *d* and some term *a*_{n} (which gives us *n*) to find *a*_{1}.