In an arithmetic series the 8th term is 34, the 12th term is 36. What is the 5th term?

A. $$31$$

B. $$32.5$$

C. $$127/4$$

D. $$251/8$$

An arithmetic series follows the following formula:

$a_n=a_1+(n-1)d$

where

$$\begin{gathered} a_n=the n^{th }term of the series \\ {a}_1=the 1st term of the series \\ n = the term position \\ d = the common different between terms \end{gathered}$$

If the 8th term is 34, then according to our equation:

$$\begin{gathered} a_n=a_1+(n-1)d \\ 34=a_1+(8-1)d \\ 34=a_1+7d (Equation \#1) \end{gathered}$$

If the 12th term is 36, then according to our equation:

$$\begin{gathered} a_n=a_1+(n-1)d \\ 36=a_1+(12-1)d \\ 36=a_1+11d (Equation \#2) \end{gathered}$$

We can now solve for a1 in terms of d by manipulating Equation #1, and then plug it into Equation #2:

$$\begin{gathered} 34=a_1+7d \\ {a}_1=34-7d \\ 36=a_1+11d \\ 36 = (34-7d)+11d \\ 36 = 34+4d \\ 2 = 4d \\ d = .5 \end{gathered}$$

Now that we know our common difference, d, is equal to .5, we can use it to solve for a1.

$$\begin{gathered} 34=a_1+7d \\ 34 = a_1+7(.5) \\ 34 = a_1+3.5 \\ 30.5 = a_1 \end{gathered}$$

Now that we have both d and a1 values, we can find the 5th term of the series:

$$\begin{gathered} a_n=30.5+(5-1)(.5) \\ {a}_n= 32.5 \end{gathered}$$

Therefore our answer is B, 32.5.