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:

53,59,527,581,5243,\frac{5}{3}, \frac{5}{9}, \frac{5}{27}, \frac{5}{81}, \frac{5}{243}, \dots

Step 1: Analyzing the sequence

The numerator remains constant at 55 across all terms. The denominator forms a geometric sequence:

3,9,27,81,243,3, 9, 27, 81, 243, \dots

This sequence is a geometric progression with: First term: 3,Common ratio: 3.\text{First term: } 3, \quad \text{Common ratio: } 3.

The nn-th term of this geometric sequence for the denominator is: 3n.3^{n}.

Step 2: General formula for ana_n

The general term ana_n can now be expressed as: an=53n.a_n = \frac{5}{3^n}.

Final Answer:

an=53n.a_n = \frac{5}{3^n}.


Would you like additional examples or further clarification? Here are 5 follow-up questions to deepen your understanding:

  1. What happens if the numerator also follows a geometric progression?
  2. How would the formula change if the sequence alternates signs, such as 53,59,527,\frac{5}{3}, -\frac{5}{9}, \frac{5}{27}, \dots?
  3. What is the sum of the first nn terms of this sequence?
  4. How would you express the nn-th term if the denominator progression starts at 99 instead of 33?
  5. 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