Math Problem Statement
Find a formula for the general term a_n of the sequence assuming the pattern of the first few terms continues: {5/3, 5/9, 5/27, 5/81, 5/243, ...}
Solution
The sequence provided is:
Step 1: Analyzing the sequence
The numerator remains constant at across all terms. The denominator forms a geometric sequence:
This sequence is a geometric progression with:
The -th term of this geometric sequence for the denominator is:
Step 2: General formula for
The general term can now be expressed as:
Final Answer:
Would you like additional examples or further clarification? Here are 5 follow-up questions to deepen your understanding:
- What happens if the numerator also follows a geometric progression?
- How would the formula change if the sequence alternates signs, such as ?
- What is the sum of the first terms of this sequence?
- How would you express the -th term if the denominator progression starts at instead of ?
- How does the concept of geometric sequences extend to real-world applications?
Tip:
When deriving formulas for sequences, always check if the terms exhibit patterns in the numerator and denominator separately!
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Math Problem Analysis
Mathematical Concepts
Sequences
Geometric Progression
Formulas
a_n = 5 / 3^n
Theorems
General term formula for geometric sequences
Suitable Grade Level
Grades 8-10