Establish Equations

Using $a_n = a + (n-1)d$, we write $a_8 = a + 7d$ and $a_{12} = a + 11d$. Their sum is $(a+7d) + (a+11d) = 2a + 18d = 100$, simplifying to $a + 9d = 50$.

$a + 9d = 50$

Use 20th Term

The 20th term is $a + 19d = 95$. We now have two linear equations: $a + 9d = 50$ and $a + 19d = 95$.

$a + 19d = 95$

Solve for Common Difference

Subtract the first equation from the second: $(a + 19d) - (a + 9d) = 95 - 50$, which yields $10d = 45$ or $d = 4.5$.

$d = 4.5$

Calculate 7th Term

First, find $a$ using $a + 9(4.5) = 50 \text{Rightarrow} a + 40.5 = 50 \text{Rightarrow} a = 9.5$. Now, calculate $a_7 = a + 6d = 9.5 + 6(4.5) = 9.5 + 27 = 36.5$.

$a_7 = 36.5$