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.