Find a_1^3/\left( {1 - 3{a_1} + 3a_1^2} \right) + a_2^3/\left( {1 - 3{a_2} + 3a_2^2} \right) + ... + a_{101}^3/\left( {1 - 3{a_{101}} + 3a_{101}^2} \right), where an = n/101.

Given the series 

\[
\sum_{n=1}^{101} \frac{a_n^3}{1 - 3a_n + 3a_n^2}
\]

where \( a_n = \frac{n}{101} \), we need to simplify the given expression.

### Step 1: Substitute \( a_n \) into the series

Substituting \( a_n = \frac{n}{101} \), we rewrite each term of the series:

\[
\frac{a_n^3}{1 - 3a_n + 3a_n^2} = \frac{\left(\frac{n}{101}\right)^3}{1 - 3\left(\frac{n}{101}\right) + 3\left(\frac{n}{101}\right)^2}.
\]

### Step 2: Simplify each term

To simplify further, let’s focus on the general term:

\[
\frac{\left(\frac{n}{101}\right)^3}{1 - 3\left(\frac{n}{101}\right) + 3\left(\frac{n}{101}\right)^2}.
\]

This simplifies to:

\[
= \frac{\frac{n^3}{101^3}}{\frac{(101 - n)^3}{101^3}}.
\]

Since both the numerator and denominator have \( 101^3 \), they cancel out:

\[
= \frac{n^3}{(101 - n)^3}.
\]

### Step 3: Recognize symmetry in the sum

Observe that:

\[
\frac{n^3}{(101 - n)^3} + \frac{(101 - n)^3}{n^3} \geq 2.
\]

### Step 4: Evaluate series 

Each \( n = 1 \) through \( n = 50 \), sums symmetrically:

\[
S = 101.
\]

Would you like further details or clarification? 

Here are related questions:
1. Can you evaluate this for different limits?
2. What other types of functions do symmetry relate?
3. Why does substitution in algebra help?
4. Can the term always reduce nicely?
5. What patterns help with symmetry? 

**Tip:** Recognize symmetry saves complex algebra!

Find a_1^3/(1 - 3{a_1} + 3a_1^2) + a_2^3/(1 - 3{a_2} + 3a_2^2) + ... + a_{101}^3/(1 - 3{a_{101}} + 3a_{101}^2), where an = n/101.

Explore the summation of a rational expression series involving terms of the form a_n^3 / (1 - 3a_n + 3a_n^2), where a_n = n/101. This problem demonstrates the use of symmetry in algebra to simplify complex summations, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation