Geometric Progression

Definition:

A geometric progression (GP) is a sequence of numbers where each term after the first is obtained by multiplying the preceding term by a fixed, non-zero number called the "common ratio."

General Form:

A GP can be represented as: a,ar,ar2,ar3,

  • a is the first term of the sequence.
  • r is the common ratio between consecutive terms.

nth Term of a GP:

The nth term of a GP is given by the formula:

an=a×r(n1)

Where:

  • an is the nth term of the GP.
  • a is the first term of the GP.
  • r is the common ratio.
  • n is the term number.

Sum of First n Terms of a GP:

The sum of the first n terms of a GP (Sn) can be calculated using the formula:

Sn=a×(1rn)1r

Where:

  • Sn is the sum of the first n terms.
  • n is the number of terms.
  • a is the first term.
  • r is the common ratio.

Properties of Geometric Progression:

  1. Common Ratio:

    • In a GP, the ratio between consecutive terms is constant.
  2. Nth Term:

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

    • The sum of the first n terms (Sn) can be calculated using Sn=a×(1rn)1r.
  4. Relationship between Terms:

    • Any term in a GP can be found by multiplying the common ratio to the previous term.
  5. Infinite GP:

    • If r<1, an infinite GP converges to a1r.
    • If r1, the series diverges.

Applications of Geometric Progression:

  • Finance: Compound interest calculations, growth and depreciation models, etc.
  • Physics: Modelling exponential growth or decay, waveforms, etc.
  • Computer Science: Algorithms involving geometric sequences, data compression, etc.