Arithmetic Progression

Definition:

An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the "common difference" and is denoted by d.

General Form:

An AP can be represented as: a,a+d,a+2d,a+3d,

  • a is the first term of the sequence.
  • d is the common difference between consecutive terms.

nth Term of an AP:

The nth term of an AP is given by the formula:

an=a+(n1)×d

Where:

  • an is the nth term of the AP.
  • a is the first term of the AP.
  • d is the common difference.
  • n is the term number.

Sum of First nTerms of an AP:

The sum of the first nterms of an AP (Sn) can be calculated using the formula:

Sn=n2×[2a+(n1)×d]

Where:

  • Sn is the sum of the first n terms.
  • n is the number of terms.
  • a is the first term.
  • d is the common difference.

Properties of Arithmetic Progression:

  1. Common Difference:

    • In an AP, the difference between consecutive terms is constant.
  2. Nth Term:

    • The formula to find the nth term of an AP is an=a+(n1)×d.
  3. Sum of an AP:

    • The sum of the first n terms (Sn) can be calculated using Sn=n2×[2a+(n1)×d].
  4. Relationship between Terms:

    • Any term in an AP can be found by adding the common difference to the previous term.
  5. Finding Number of Terms:

    • n=anad+1 can be used to find the number of terms in the AP.

Applications of Arithmetic Progression:

  • Mathematics: Formulas for arithmetic means, linear interpolation, etc.
  • Finance: Calculating depreciation, amortization, etc.
  • Physics: Modelling uniformly changing quantities like distance, velocity, etc.
  • Computer Science: Algorithms, series in data structures, etc.