Question

How do you find the $n^{th}$ term rule for $1,5,9,13,...$?

Answer 1

Solve for the ${\text{n}}^{th}$term rule:

Given : $1,5,9,13,...$

Here the difference between the terms is constant;

$5-1=4$

$$\begin{gathered} 9-5=4 \\ 13-9=4 \end{gathered}$$

Therefore sequence is Arithmetic sequence.

Thus, the common difference $\mathrm{d}=4$ and the first term of the series is $\mathrm{a}=1$

Now, the $n^{th}$term of the series is

${\mathrm{a}}_{\mathrm{n}}=\mathrm{a}+\left(\mathrm{n}-1\right)\mathrm{d}$

$\Rightarrow$${\mathrm{a}}_{\mathrm{n}}=1+\left(\mathrm{n}-1\right)\left(4\right)$

$\Rightarrow$${\mathrm{a}}_{\mathrm{n}}=1+4\mathrm{n}-4$

$\Rightarrow$${\mathrm{a}}_{\mathrm{n}}=4\mathrm{n}-3$

Hence, the required $n^{th}$term rule is ${\mathrm{a}}_{\mathrm{n}}=4\mathrm{n}-3$.