What Have You Learned About Geometric Sequences?

I've learned that a geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a constant value, known as the common ratio.

Key Characteristics of Geometric Sequences

Here's a breakdown of what I understand about geometric sequences:

• Definition: A sequence where the ratio between consecutive terms is constant.
• Common Ratio (r): The constant factor multiplied to each term to get the next. To find the common ratio, you divide any term by its preceding term.
• General Form: a, ar, ar², ar³, ... , where 'a' is the first term.
• nth Term Formula: a_n = a * r^(n-1), where:
  • a_n is the nth term
  • a is the first term
  • r is the common ratio
  • n is the term number
• Identifying Geometric Sequences: To determine if a sequence is geometric, divide each term by the preceding term. If the result is the same constant value for all pairs of consecutive terms, then it's a geometric sequence.

Example

Consider the sequence: 2, 6, 18, 54,...

1. Check the Ratio:

   • 6 / 2 = 3
   • 18 / 6 = 3
   • 54 / 18 = 3

2. Conclusion: Since the ratio is consistently 3, this is a geometric sequence with a common ratio (r) of 3 and a first term (a) of 2.

3. Finding the 5th Term: Using the formula a_n = a * r^(n-1), we can find the 5th term (a₅):

   • a₅ = 2 * 3^(5-1)
   • a₅ = 2 * 3⁴
   • a₅ = 2 * 81
   • a₅ = 162

Summary

In essence, a geometric sequence is a series of numbers that follow a consistent multiplicative pattern, defined by its first term and its common ratio. We can use formulas to find any term in the sequence, given the first term, common ratio, and term position.