Question

How do you write a rule for the nth term of the arithmetic sequence and then find `a_22` given `45, 31, 17, 3,...`?

Answer 1

`a_n = 45 - (14) xx n`
`a_22 = -263`

Explanation:

First, we figure out the sequence. Then we generalize it.
This one shows a decrease of 14 (-14) between every value in the series.

So, if we start with `a_0 = 45` we see that `a_1 = a_0 - 14`.
The next one would be `a_2 = a_1 - 14`. Substituting our value for `a_1` from the previous difference we see that:

`a_2 = (a_0 - 14) - 14` or `a_2 = a_0 - (14) xx 2`

Recognizing that now our multiplier is the same as our sequence identifier (2) we can generalize the expression to:

`a_n = a_0 - (14) xx n`; or `a_n = 45 - (14) xx n`

NOW putting in our values for `a_22` we can obtain:
`a_22 = a_0 - (14) xx 22` ; `a_22 = 45 - 308 = -263`