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`).