Arithmetic Progression

Definition

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term.

Key Features

• Common difference (d): the fixed constant added to each term to get the next term
• First term (a): the starting point of the sequence
• Formula: an = a + (n - 1)d, where an is the nth term

Examples

• 2, 5, 8, 11, ... (d = 3, a = 2)
• 10, 15, 20, 25, ... (d = 5, a = 10)

Properties

• The sequence increases or decreases by a fixed amount (d) at each step
• The nth term can be calculated using the formula an = a + (n - 1)d
• The sum of the first n terms (Sn) can be calculated using the formula Sn = (n/2)(2a + (n - 1)d)

Applications

• Modeling real-world problems involving uniform change, such as population growth, financial transactions, and motion
• Solving problems involving sequences and series
• Used in various fields, including mathematics, physics, engineering, and economics

Arithmetic Progression (AP)

• A sequence of numbers where each term after the first is obtained by adding a fixed constant (common difference) to the previous term

Key Features of AP

• Common difference (d): the fixed constant added to each term to get the next term
• First term (a): the starting point of the sequence
• Formula: an = a + (n - 1)d, where an is the nth term

Examples of AP

• 2, 5, 8, 11,... (d = 3, a = 2)
• 10, 15, 20, 25,... (d = 5, a = 10)

Properties of AP

• The sequence increases or decreases by a fixed amount (d) at each step
• The nth term can be calculated using the formula an = a + (n - 1)d
• The sum of the first n terms (Sn) can be calculated using the formula Sn = (n/2)(2a + (n - 1)d)

Applications of AP

• Used to model real-world problems involving uniform change, such as:
  • Population growth
  • Financial transactions
  • Motion
• Used to solve problems involving sequences and series
• Applications in various fields, including:
  • Mathematics
  • Physics
  • Engineering
  • Economics

Arithmetic Progression

Formula

• An arithmetic progression is a sequence of numbers where each term is obtained by adding a fixed constant to the previous term.
• The formula to find the nth term is: an = a1 + (n - 1) * d
• In this formula: an is the nth term, a1 is the first term, n is the term number, and d is the common difference.

Sum

• The formula to find the sum of the first n terms is: Sn = (n/2) * (a1 + an)
• In this formula: Sn is the sum of the first n terms, a1 is the first term, and an is the nth term.

Nth Term

• The nth term can be found using the formula: an = a1 + (n - 1) * d
• Alternatively, the nth term can be found using the formula: an = a(n-1) + d
• In both formulas: an is the nth term, a1 is the first term, n is the term number, and d is the common difference.