Finding the nth term of an arithmetic sequence is easy once you understand it!

What is an Arithmetic Sequence?

An arithmetic sequence is a list of numbers where the difference between each number and the next one stays the same. This steady difference is called the common difference, and we can show it as (d).

Step 1: Know the Formula

To find the nth term ((T_n)) of an arithmetic sequence, you can use this formula:

[ T_n = a + (n - 1)d ]

Here’s what the symbols mean:

• (T_n) is the term you want to find.
• (a) is the first term of the sequence.
• (n) is the number of the term you want.
• (d) is the common difference.

Step 2: Find the First Term and Common Difference

Let’s use this sequence as an example: 3, 7, 11, 15, ...

• First term ((a)) = 3
• Common difference ((d)) = 7 - 3 = 4

Step 3: Use the Formula

If you want to find the 5th term ((n = 5)), plug the numbers into the formula:

[ T_5 = 3 + (5 - 1) \cdot 4 ]

Now, solve it step by step:

[ T_5 = 3 + 4 \cdot 4 ]

[ T_5 = 3 + 16 ]

[ T_5 = 19 ]

Example

What if you need the 10th term ((n = 10))?

Use the formula again:

[ T_{10} = 3 + (10 - 1) \cdot 4 ]

Now solve this too:

[ T_{10} = 3 + 36 ]

[ T_{10} = 39 ]

And that’s it! Just remember the formula, find your first term and your common difference, and you can find any term in the sequence!