Question

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

Answer 1

Explanation to find the first term of an arithmetic sequence:

If we have two terms given we could use the $n^{th}$ term formula to calculate the first term.

Example: If the second term of an arithmetic sequence is $4$ and fifth term is $10$ then 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 i.e. $a_n=a_1+\left(n-1\right)d$ to build a system of equations; where $a_1$ is the first term, $d$ is the common difference

$$\begin{gathered} \therefore a_2=4 \\ \Rightarrow a_1+\left(2-1\right)d=4 \\ \Rightarrow a_1+d=4...\left(i\right) \end{gathered}$$

And, $a_5=10$

$\Rightarrow$$a_1+\left(5-1\right)d=10$

$\Rightarrow$ $a_1+4d=10...\left(ii\right)$

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

$\Rightarrow$$\left[a_1+4d\right]-\left[a_1+d\right]=10-4$

$\Rightarrow$ $3d=6$

$\Rightarrow$ $d=2$

Now if we substitute the value of $d$ in equation $\left(i\right)$ we get $a_1+2=4\Rightarrow a_1=2.$

Now we can answer that the first term of this sequence is $2$.

Hence, finding the first term of an arithmetic sequence is explained.