# How do you tell whether the sequence with the terms a_3=21, a_5=27, a_12=48 is arithmetic?

Mar 9, 2017

It is an arithmetic series. See details below.

#### Explanation:

In an arithmetic series we can find common difference between two terms ${a}_{m}$ and ${a}_{n}$ by using formula $d = \frac{{a}_{n} - {a}_{m}}{n - m}$, if common difference is same throughout, we can say this is an arithmetic sequence.

Here we can get three values of $d$

1. $\frac{{a}_{5} - {a}_{3}}{5 - 3} = \frac{27 - 21}{2} = \frac{6}{2} = 3$

2. $\frac{{a}_{12} - {a}_{5}}{12 - 5} = \frac{48 - 27}{7} = \frac{21}{7} = 3$ and

3. $\frac{{a}_{12} - {a}_{3}}{12 - 3} = \frac{48 - 21}{9} = \frac{27}{3} = 3$

As we get $d$ as constantly same, it is an arithmetic series.