What is a geometric sequence?

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Here's a breakdown:

• Definition: A sequence where the ratio between successive terms is constant.

• Formula: The general form of a geometric sequence is:

  • a, ar, ar², ar³, ...
  • Where:
    • a is the first term.
    • r is the common ratio.

• Finding the Common Ratio: Divide any term by its preceding term.

• nth Term: The nth term (a_n) of a geometric sequence is given by: a_n = ar^n-1

• Sum of a Finite Geometric Series: The sum (S_n) of the first n terms of a geometric series is:

  • S_n = a(1 - rⁿ) / (1 - r) if r ≠ 1

• Sum of an Infinite Geometric Series: If the absolute value of the common ratio is less than 1 (|r| < 1), the sum of an infinite geometric series converges to:

  • S = a / (1 - r)

• Examples:

  • 2, 4, 8, 16, ... (a = 2, r = 2)
  • 10, 5, 2.5, 1.25, ... (a = 10, r = 0.5)