**Arithmetic Progressions (AP): A Comprehensive Guide**

**Introduction:**

In mathematics, an **arithmetic progression (AP)**, also known as an arithmetic sequence, is a special type of ordered list where the difference between consecutive terms remains constant. This fixed difference is called the **common difference**, denoted by `d`

.

**General Form:**

An AP can be expressed in two common ways:

**Explicit Formula:**Each term is represented by a specific formula derived from the first term (`a_1`

) and the common difference (`d`

). The`n`

th term is given by:`a_n = a_1 + (n - 1)d`

**Recursive Formula:**This approach expresses each term based on the previous term(s). In an AP, the recursive formula takes the form:`a_n = a_{n-1} + d`

where `a_1`

is the first term and `d`

is the common difference.

**Examples:**

- The sequence 2, 5, 8, 11, 14, … is an AP with
`a_1 = 2`

and`d = 3`

. - The sequence 7, 4, 1, -2, -5, … is an AP with
`a_1 = 7`

and`d = -3`

.

**Properties:**

- The sum of any two consecutive terms in an AP is equal to the sum of the first and last terms.
- The average of any
`n`

consecutive terms in an AP is equal to the average of the first and last terms. - An AP can be finite (having a specific number of terms) or infinite (continuing indefinitely).

**Applications:**

Arithmetic progressions have numerous applications in various fields, including:

**Finance:**Calculating compound interest, loan payments, and annuity values.**Physics:**Determining the displacement of objects in uniform motion with acceleration.**Statistics:**Studying the distribution of data in frequency distributions.**Computer Science:**Analyzing the complexity of algorithms.

**Sum of n Terms:**

The sum of `n`

terms in an AP, denoted by `S_n`

, is given by the formula:`S_n = n/2 [2a_1 + (n - 1)d]`

This formula is derived from the concept of an arithmetic series, which is the sum of all terms in an AP. By pairing the first and last terms, second and second-last terms, and so on, we can show that the sum of an AP is equal to the number of terms (`n`

) multiplied by the average of the first and last terms.

**Practical Applications:**

**Calculating the total distance traveled by a train:**If a train starts at rest and accelerates uniformly to a final speed, its distance traveled after`n`

seconds can be modeled by an AP. The first term represents the initial distance, and the common difference represents the constant acceleration.**Finding the total cost of renting a car:**If the daily rental cost of a car increases by a fixed amount each day, the total cost for`n`

days can be calculated using the sum of an AP.**Estimating the total population of a city:**If a city’s population increases at a constant rate each year, the total population after`n`

years can be approximated using the sum of an AP.

**Example:**

A car rental company charges $50 on the first day and increases the daily charge by $10 each day. What is the total cost of renting the car for 5 days?

Solution:

- This is an arithmetic progression with
`a_1 = 50`

and`d = 10`

. - We need to find
`S_5`

, the sum of 5 terms. - Using the formula
`S_n = n/2 [2a_1 + (n - 1)d]`

, we get:

`S_5 = 5/2 [2 * 50 + (5 - 1) * 10] = 5/2 * 130 = 325`

Therefore, the total cost of renting the car for 5 days is $325.

**Remember:**

- An AP is characterized by a constant difference between consecutive terms.
- The explicit and recursive formulas can be used to find any term in the sequence.
- The sum of
`n`

terms is calculated using the`S_n`

formula, which involves the first term, last term