The fifth term of an arithmetic sequence is 9 and the 32nd term is -84. What is the 23rd term?

Let a be the first term, and r be the rate of the arithmetic sequence. We have:

 

a + 4r = 9

a + 31r = -84

a + 22 r = ?

Realize that we can Subtract our first equation from our second equation. This gives us:

 

27r = -93

r = -93/27

 

since we already have that a+31r = -84, we can just subtract 9r from this equation to get:
 

a + 22r = -84 -(9*-93/27) = -84 -(-31) = -84 + 31 = -53

It looks like a standard question that you could find in any school.

There is nothing unusual about this question.