# Calculation of the first common term of two arithmetic sequences

Consider the two arithmetic sequences:

$$\{-3, 4, 11, ...\},\ \{2, 7, 12, ...\}$$

If we continue the terms of the two sequences, the first common term of the two sequences is 32.

But the question is, how to calculate the first common term of two sequences without continuing and only with mathematical calculations and for example using the general term of two sequences?

You can find the expression for the general term of each sequence and equate them.

In this specific case, the first has general term $$-3 + 7(m-1) = 7m-10$$, while the second, $$2 + 5(n-1) = 5n-3$$

Equate, $$7m-10 = 5n-3$$

$$7m - 7 = 5n$$

Since the left hand side is a multiple of $$7$$ and $$5$$ and $$7$$ are coprime, we need $$n = 7k$$, where $$k$$ is some integer.

So, after substitution and division throughout by $$7$$, we get $$m = 5k + 1$$.

Hence concurrence between the two sequences occurs for $$(5k+1,7k)$$ where $$k$$ can take any positive integer value.

For instance, $$k=1$$ gives you the case of $$(6,7)$$ or the term $$32$$, which you have already discovered.