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

Sep 22, 2017

${a}_{n} = 45 - \left(14\right) \times n$
${a}_{22} = - 263$

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:

${a}_{2} = \left({a}_{0} - 14\right) - 14$ or ${a}_{2} = {a}_{0} - \left(14\right) \times 2$

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

${a}_{n} = {a}_{0} - \left(14\right) \times n$; or ${a}_{n} = 45 - \left(14\right) \times n$

NOW putting in our values for ${a}_{22}$ we can obtain:
${a}_{22} = {a}_{0} - \left(14\right) \times 22$ ; ${a}_{22} = 45 - 308 = - 263$