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 4 of 4: Count the terms and divide by two

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In plain words

Don't forget that the index runs from $0$ to $101$ inclusive, so there are $101 - 0 + 1 = 102$ terms in total, making the average sum $51$ .

2S = \sum_{i=0}^{101} 1 = 102 \implies S = 51

Detailed analysis

Since $x_i^3 + x_{101-i}^3 > 0$ for every $i \in \{0, 1, \ldots, 101\}$ , each of the $102$ terms in the combined sum equals $1$ . Therefore $2S = 102$ , which gives $S = 51$ .