# How do I find the first term of an arithmetic sequence?

How do I find the first term of an arithmetic sequence?
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brodireo1
To answer this (or any other) question I need some data. It is impossible to solve such task without having anything given.
If I had two terms I could use the n−th term formula to calculate the first term.
For example:
The second term of an arithmetic sequence is 4. The fifth is 10. Find the first term.
In this task we have 2 terms given: ${a}_{2}=4$ and ${a}_{5}=10$. We can use the n−th term formula to build a system of equations:
$\left\{\begin{array}{l}{a}_{1}+d=4\\ {a}_{1}+4d=10\end{array}$
If we subtract the first equation from the second we can calculate d:
3d=6; d=2
Now if we substitute the calculated value we see that: ${a}_{1}+2=4$, so ${a}_{1}=2$.
Now we can answer that The first term of this sequence is 2