**What is an Arithmetic Sequence?**

A ** sequence** is list of numbers where the same operation(s) is done to one number in order to get the next.

**specifically refer to sequences constructed by adding or subtracting a value – called the**

**Arithmetic sequences****to get the next term.**

**common difference**–In order to efficiently talk about a sequence, we use a formula that builds the sequence when a list of indices are put in. Typically, these formulas are given one-letter names, followed by a parameter in parentheses, and the expression that builds the sequence on the right hand side.

`a(n) = n + 1`

Above is an example of a formula for an arithmetic sequence.

**Examples**

Sequence: 1, 2, 3, 4, … | Formula: a(n) = n + 13

Sequence: 8, 13, 18, … | Formula: b(n) = 5n - 2

**A Recursive Formula**

Note: Mathematicians start counting at 1, so by convention, `n=1`

is the first term. So we must define what the first term is. Then we have to figure out and include the common difference.

Taking a look at the examples again,

Sequence: 1, 2, 3, 4, … | Formula: a(n) = n + 1 | Recursive formula: a(n) = a(n-1) + 1, a(1) = 1

Sequence: 3, 8, 13, 18, … |Formula: b(n) = 5n - 2 | Recursive formula: b(n) = b(n-1) + 5, b(1) = 3

**Finding the Formula (given a sequence with the first term)**

```
1. Figure out the common difference
Pick a term in the sequence and subtract the term that comes before it.
2. Construct the formula
The formula has the form: `a(n) = a(n-1) + [common difference], a(1) = [first term]`
```

**Finding the Formula (given a sequence without the first term)**

```
1. Figure out the common difference
Pick a term in the sequence and subtract the term that comes before it.
2. Find the first term
i. Pick a term in the sequence, call it `k` and call its index `h`
ii. first term = k - (h-1)*(common difference)
3. Construct the formula
The formula has the form: `a(n) = a(n-1) + [common difference], a(1) = [first term]`
```

**More Information:**

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