Problem 1

Compute the sum

S = \sum_{i=0}^{101} \frac{x_i^3}{1 - 3x_i + 3x_i^2}

for

x_i = \frac{i}{101}

Step 1 of 4: Identify the complementary relationship between terms

Previous step Next step

In plain words

The points $x_i$ divide the interval $[0, 1]$ into $101$ equal parts, so reflecting across the midpoint $\frac{1}{2}$ pairs $x_i$ with $1 - x_i = x_{101-i}$ .

x_{101-i} = \frac{101-i}{101} = 1 - \frac{i}{101} = 1 - x_i

Detailed analysis

For each index $i \in \{0, 1, \ldots, 101\}$ , the reflected index $101 - i$ also lies in $\{0, 1, \ldots, 101\}$ , and the corresponding nodes satisfy $x_{101-i} = \frac{101-i}{101} = 1 - x_i$ .