Math Problem Statement

Express the sum 2/6 + 2/9 + 2/12 + 2/15 + ... + 2/24 in sigma notation.

Solution

The problem asks us to express the given sum in sigma notation. Let's analyze the terms:

The sequence of terms is:
26,29,212,215,,224\frac{2}{6}, \frac{2}{9}, \frac{2}{12}, \frac{2}{15}, \dots, \frac{2}{24}

Step 1: Factorize the denominators

  • 26=223\frac{2}{6} = \frac{2}{2 \cdot 3}
  • 29=232\frac{2}{9} = \frac{2}{3^2}
  • 212=2223\frac{2}{12} = \frac{2}{2^2 \cdot 3}
  • 215=235\frac{2}{15} = \frac{2}{3 \cdot 5}
  • \dots

Step 2: Identify the general term

The denominators are the products of integers 23,32,223,2 \cdot 3, 3^2, 2^2 \cdot 3, \dots. This suggests the terms are related to the integers nn in some way.

The pattern shows that the denominators are multiples of 3, with powers alternating between 2 and other integers.

Step 3: Construct sigma notation

The numerator is always 2, and the denominator follows a sequence based on nn. From the structure of the denominators, the sum can be written as:

n=1N23n\sum_{n=1}^N \frac{2}{3n}

where NN is the upper limit.

Would you like to clarify this or discuss further?


Follow-up Questions:

  1. What are the steps to analyze and find patterns in sequences?
  2. Can this sum be simplified or rewritten in another way?
  3. How do the limits of summation affect the outcome?
  4. Could this series be part of a larger mathematical context (e.g., harmonic series)?
  5. How would changing the numerator affect the sigma notation?

Tip:

When constructing sigma notation, always look for consistent patterns in numerators and denominators first!

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Math Problem Analysis

Mathematical Concepts

Series and Summation
Patterns in Sequences
Sigma Notation

Formulas

General term of a series
Summation formula

Theorems

Arithmetic and Geometric Progression

Suitable Grade Level

Grades 9-11