In an arithmetic sequence with 20 terms, the sum of the first three terms is 15 and the sum of the last three terms is 12 . What is the sum of the 20 terms in the sequence?

(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, $3,5,7,9$ are the first four terms of an arithmetic sequence.)
Suppose that the sequence has first term $a$ and common difference $d$.

Then the 20 terms are $a, a+d, a+2 d, \ldots, a+17 d, a+18 d, a+19 d$.

The sum of an arithmetic sequence with first term $a$, common difference $d$, and $n$ terms is $\frac{n}{2}\left(a+(a+(n-1) d)\right.$, so the sum of this sequence is $\frac{20}{2}(a+(a+19 d))=10(2 a+19 d)$.

Since the sum of the first three terms is 15 , then $a+(a+d)+(a+2 d)=15$ or $3 a+3 d=15$ and so $a+d=5$.

Since the sum of the last three terms is 12 , then $(a+17 d)+(a+18 d)+(a+19 d)=12$ or $3 a+54 d=12$ and so $a+18 d=4$.

Therefore, the sum of all 20 terms is $10(2 a+19 d)=10((a+d)+(a+18 d))=10(5+4)=90$.