Geometric Sequences Overview

Geometric Sequences Overview

• Explicit formula for a geometric sequence is used to calculate the nth term directly.
• Recursive formula for a geometric sequence calculates the next term based on the previous term.

Common Ratio

• The common ratio (r) is the factor by which each term is multiplied to obtain the next term in the sequence.
• Example provided: For the sequence 6, -3, (3/2), -(3/4), the common ratio is r = -(1/2).

Terms Calculation

• To find specific terms in a geometric sequence, use the formulas derived from the initial term and the common ratio.
• For the sequence 2, 6, 18, 54, the 7th term is 1458.
• For the sequence 3, 9, 27, the 8th term is 6561.

Formulas

• The common ratio for a given formula like a = 5*(3)^(n-1) is r = 3.
• Explicit formula for the sequence 1, -(1/2), (1/4), -(1/8) is a = 1*(-1/2)^(n-1).
• Explicit formula for the sequence 2, 6, 18, 54 is a = 2*(3)^(n-1).

Graphical Representation

• The graph of a geometric sequence is characterized by an exponential curve, illustrating rapid growth or decay.

Purpose of Formulas

• The explicit formula is utilized to determine the nth term or to find any specific term in a geometric sequence.
• The recursive formula is advantageous for calculating the next term based on the most recent term in the sequence.