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

Oct 23, 2016

See explanation.

#### Explanation:

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 $n - t h$ 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 $n - t h$ term formula to build a system of equations:

$\left\{\begin{matrix}{a}_{1} + d = 4 \\ {a}_{1} + 4 d = 10\end{matrix}\right.$

If we subtract the first equation from the second we can calculate $d$:

$3 d = 6$; $d = 2$

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