# Tell whether the sequence with the terms a_3=21,a_5=27,a_12=48 is arithmetic

Tell whether the sequence with the terms ${a}_{3}=21,{a}_{5}=27,{a}_{12}=48$ is arithmetic
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takwerkyo0
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.