Geometric Sequences Quiz

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 (r).
• The general form of a geometric sequence can be expressed as a, ar, ar², ar³, ..., where 'a' is the first term.

Differentiating Geometric and Arithmetic Sequences

• In a geometric sequence, the ratio between consecutive terms is constant (common ratio), while in an arithmetic sequence, the difference between consecutive terms is constant (common difference).
• Example: In the geometric sequence 2, 6, 18, the common ratio is 3 (6/2 = 3, 18/6 = 3). In the arithmetic sequence 2, 4, 6, the common difference is 2 (4-2 = 2, 6-4 = 2).

nth Term of a Geometric Sequence

• The nth term (Tn) of a geometric sequence can be calculated using the formula: Tn = a * r^(n-1), where 'a' is the first term, 'r' is the common ratio, and 'n' is the term number.

Geometric Means

• Geometric means are terms that can be inserted between two known terms of a geometric sequence to maintain the ratio.
• For two terms 'a' and 'b', the geometric mean (GM) can be found using the formula: GM = √(a*b).

Sum of the Terms of a Geometric Sequence

• The sum of the first n terms (S_n) of a geometric sequence can be calculated using the formula:
  • If r ≠ 1: S_n = a * (1 - r^n) / (1 - r)
  • If r = 1: S_n = n * a, where 'n' is the number of terms.
• For an infinite geometric series where |r| < 1, the sum can be calculated using: S = a / (1 - r).