Arithmetic Progression Formula

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.
• The fixed constant is called the common difference (d).

Formula

• The general formula for an arithmetic progression is: an = a1 + (n - 1)d where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.

Key Properties

• Common difference (d) is the same throughout the sequence.
• The sequence progresses by adding the common difference to the previous term.
• The nth term can be calculated using the formula an = a1 + (n - 1)d.

Examples

• 2, 5, 8, 11, 14... (a1 = 2, d = 3)
• 10, 15, 20, 25, 30... (a1 = 10, d = 5)

Applications

• Calculating the sum of an arithmetic series.
• Modeling real-world problems, such as:
  • Distance traveled by an object moving at a constant acceleration.
  • Increase in population size over time.
  • Compound interest on an investment.

Important Concepts

• Arithmetic mean: The average of two or more numbers in an arithmetic progression.
• Arithmetic series: The sum of a finite number of terms in an arithmetic progression.
• Infinite arithmetic progression: A sequence that extends indefinitely in one direction.

Definition

• An arithmetic progression (AP) is a sequence of numbers where each term is obtained by adding a fixed constant, called the common difference (d), to the previous term.

Formula

• The general formula for an arithmetic progression is: an = a1 + (n - 1)d, where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.

Key Properties

• The common difference (d) remains the same throughout the sequence, allowing the sequence to progress by adding the common difference to the previous term.
• The nth term can be calculated using the formula an = a1 + (n - 1)d.

Examples

• The sequence 2, 5, 8, 11, 14... has a first term (a1) of 2 and a common difference (d) of 3.
• The sequence 10, 15, 20, 25, 30... has a first term (a1) of 10 and a common difference (d) of 5.

Applications

• Arithmetic progressions can be used to calculate the sum of an arithmetic series.
• APs can model real-world problems, such as:
  • Distance traveled by an object moving at a constant acceleration.
  • Increase in population size over time.
  • Compound interest on an investment.

Important Concepts

• Arithmetic mean: The average of two or more numbers in an arithmetic progression.
• Arithmetic series: The sum of a finite number of terms in an arithmetic progression.
• Infinite arithmetic progression: A sequence that extends indefinitely in one direction.