Question

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

Answer 1

In the given question, clearly, we have been given an AP. We have been asked to find the position of any term from the start, i.e., the ${n^{th}}$ term. For doing that, first we are going to write the first term and common difference of the AP. Then we are going to write the formula of ${a_n}$ and put in the values for this term.

Formula used:
We are going to use the formula of ${a_n}$, which is:
${a_n} = a + \left( {n - 1} \right)d$

Complete step-by-step answer:
The formula for ${a_n}$ is:
${a_n} = a + \left( {n - 1} \right)d$
Here, first term, $a = 1$
common difference, second term minus first term, $d = 5 - 1 = 4$
${a_n} = 1 + \left( {n - 1} \right)4$
Opening the brackets,
${a_n} = 1 + 4n - 4 = 4n - 3$
Therefore, the rule for ${n^{th}}$ term for this AP is “$4n - 3$”.

Note: In the given question, we were given an arithmetic progression. We had to write the formula of the ${n^{th}}$ term. To do that, we write the formula of ${n^{th}}$term, put in the values and simplify the answer. We only need to pay attention to the formula of the ${n^{th}}$term. It is the only thing around which the answer revolves.