When you're learning about arithmetic sequences, it helps to know the difference between two types of formulas: explicit and recursive. Understanding these can make working with sequences a lot easier.

Explicit Formula

The explicit formula is a straightforward way to find the $n^{th}$ term in a sequence. You don't need to know the previous terms to find it. It looks like this:

$$a_n = a_1 + (n - 1)d$$

In this formula:

• $a_n$ is the $n^{th}$ term you want to find.
• $a_1$ is the first term in the sequence.
• $d$ is the common difference between the terms.
• $n$ is the number of the term you want.

So, if you know the first term and the common difference, you can easily find any term you need. This is really helpful!

For example, let’s say the first term is 3 (that’s $a_1 = 3$) and the common difference is 2. If you want to find the 10th term, you just plug in the numbers:

$$a_{10} = 3 + (10 - 1) \cdot 2 = 3 + 18 = 21$$

So, the 10th term is 21!

Recursive Formula

Now, the recursive formula is a bit different. This formula helps you find each term based on the one before it. It typically looks like this:

$$a_n = a_{n-1} + d$$

You also need to state the first term like this:

$$a_1 = \text{(first term)}$$

With this formula, you start with the first term and keep adding the common difference to get the next terms. Using the same example, if $a_1 = 3$ and $d = 2$, here’s how you’d find the terms:

To find $a_2$, you add the common difference to the first term:

$$a_2 = a_1 + d = 3 + 2 = 5$$

Then, to find $a_3$, you do:

$$a_3 = a_2 + d = 5 + 2 = 7$$

Summary

In short, the explicit formula is great for quick calculations and lets you find any term right away. The recursive formula helps you build the sequence step by step from the previous terms. Both formulas are useful, depending on how you want to work with sequences!