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

Question

brodireo1 · Accepted Answer

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:

{\begin{cases} a_{1} + d = 4 \\ a_{1} + 4 d = 10 \end{cases}

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

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

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