Question

A series contains 18 numbers and has a sum of 4,185. The last number of the series is 275. What is the first number?

Answer 1

If the series is made up of numbers in an arithmetic progression, you have

$\displaystyle\sum_{n=1}^{18}a_n=\sum_{n=1}^{18}(a_1+(n-1)d)=18a_1+153d=4185$

where $a_1$ is the first term and $d$ is the common difference between terms.

Since the last term in the series is $a_{18}=275$, you have

$a_{18}=a_1+(18-1)d\implies a_1+17d=275$

Solve the system

$\begin{cases}18a_1+153d=4185\\a_1+17d=275\end{cases}$

and you'll find that the first term is $a_1=190$ (with $d=5$).