An arithmetic series contains 20 numbers. The first number is 102. The last number is 159. Which expression represents the sum of the series?

If [tex]a_1=102[/tex] is the first term in the series, then the next term is [tex]a_2=a_1+k[/tex] for some fixed [tex]k[/tex], the term after that is [tex]a_3=a_2+k=a_1+2k[/tex], and so on. So the 20th term in the series would be [tex]a_{20}=a_1+(20-1)k=102+19k=159[/tex].

We have enough info to find [tex]k[/tex]:

[tex]102+19k=159\implies19k=57\implies k=3[/tex]

So the sum of the series is given by

[tex]\displaystyle\sum_{n=1}^{20}(a_1+3(n-1))=\sum_{n=1}^{20}(3n+99)=2610[/tex]