## Arithmetic Sequence

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**.

### Key Concepts:

**General Form:**An arithmetic sequence can be represented by the general form:**a, a + d, a + 2d, a + 3d, ...**where:**a**is the first term**d**is the common difference

**nth Term:**The**nth term**of an arithmetic sequence is given by the formula:**a<sub>n</sub> = a + (n - 1)d****Sum of n terms:**The sum of the first**n**terms of an arithmetic sequence is given by the formula:**S<sub>n</sub> = n/2 [2a + (n - 1)d]**

### Examples:

**Example 1:**

The sequence **2, 5, 8, 11, 14...** is an arithmetic sequence because the common difference is **3**.

**a = 2****d = 3**

**Example 2:**

Find the 10th term of the arithmetic sequence **3, 7, 11, 15...**

**a = 3****d = 4****n = 10**

Using the formula **a<sub>n</sub> = a + (n - 1)d**, we get:
a<sub>10</sub> = 3 + (10 - 1)4 = 3 + 36 = **39**

**Example 3:**

Calculate the sum of the first 20 terms of the arithmetic sequence **1, 4, 7, 10...**

**a = 1****d = 3****n = 20**

Using the formula **S<sub>n</sub> = n/2 [2a + (n - 1)d]**, we get:
S<sub>20</sub> = 20/2 [2(1) + (20 - 1)3] = 10 [2 + 57] = **590**

### Applications:

Arithmetic sequences have various applications in mathematics and real-world scenarios, including:

**Linear functions:**The relationship between consecutive terms in an arithmetic sequence is linear.**Financial calculations:**Calculating compound interest, loan repayments, and other financial computations can involve arithmetic sequences.**Physics:**Analyzing motion with constant acceleration and other physical phenomena can utilize arithmetic sequences.

### Conclusion:

Arithmetic sequences are a fundamental concept in mathematics that provides a framework for understanding sequences with a constant difference between terms. They have numerous applications in various fields, making them an essential part of mathematical knowledge.