The nth term

The $n$th term of an arithmetic sequence gives a formula to find any term without listing them all. Instead of using the first term ($a_1$), we can use the common difference ($d$) and the 0th term ($a_0$). The formula is:

$$a_n=d\cdot n+a_0$$

Here, $d$ is the common difference, and $a_0$ is the term before the first term (when $n=0$).

Example: Find the $n$th Term

Consider the sequence:

$$5,8,11,14,\dots$$

Step 1: Find the common difference ($d$): Subtract consecutive terms. Here, $d=8-5=3$.

Step 2: Find the 0th term ($a_0$): Subtract the common difference from the first term. Here:

$$a_0=5-3=2$$

Step 3: Write the formula:

The formula becomes:

$$a_n=3n+2$$

Using the $n$th Term Formula

You can use this formula to find any term. For example, to find the 7th term ($a_7$):

$$a_7=3(7)+2=21+2=23$$

The 7th term is 23.

Summary

• Find the common difference ($d$) by subtracting consecutive terms.
• Find the 0th term ($a_0$) by subtracting $d$ from the first term.
• Use the formula $a_n=d\cdot n+a_0$ to find any term in the sequence.

Using the common difference and 0th term makes it easier to work with arithmetic sequences.