## Arithmetic Sequence Formula

An arithmetic sequence is a sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the **common difference**.

### Formula for the nth term of an arithmetic sequence

The formula for the nth term of an arithmetic sequence is:

**a<sub>n</sub> = a<sub>1</sub> + (n - 1)d**

where:

**a<sub>n</sub>**is the nth term**a<sub>1</sub>**is the first term**n**is the number of the term**d**is the common difference

### Example

Let's say we have an arithmetic sequence with the first term (a<sub>1</sub>) being 3 and a common difference (d) of 2. To find the 5th term (a<sub>5</sub>), we can use the formula:

a<sub>5</sub> = a<sub>1</sub> + (5 - 1)d

a<sub>5</sub> = 3 + (5 - 1)2

a<sub>5</sub> = 3 + 8

**a<sub>5</sub> = 11**

Therefore, the 5th term of this arithmetic sequence is 11.

### Applications of arithmetic sequences

Arithmetic sequences have many applications in real life, such as:

**Calculating compound interest:**The amount of money you earn on a fixed deposit increases by a fixed amount each year, forming an arithmetic sequence.**Analyzing linear growth:**When a quantity increases at a constant rate, it follows an arithmetic sequence.**Predicting patterns:**Arithmetic sequences can be used to predict patterns in data, such as the number of customers entering a store at different times of day.

### Other formulas

Besides the formula for the nth term, there are other useful formulas for arithmetic sequences:

**Sum of the first n terms:**S<sub>n</sub> = n/2 * (a<sub>1</sub> + a<sub>n</sub>)**Sum of the first n terms (alternative form):**S<sub>n</sub> = n/2 * [2a<sub>1</sub> + (n - 1)d]

These formulas can be used to calculate the sum of a specific number of terms in an arithmetic sequence.