Evaluate the sum 1/(1×2) + 1/(2×3) + 1/(3×4) + … + 1/(9×10). Give it as a fraction in lowest terms.

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Evaluate the sum $\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \dots + \frac{1}{9 \times 10}$ . Give it as a fraction in lowest terms.

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Hint

Write each term as a difference of $2$ simple fractions.

Solution

Each term splits: $\frac{1}{k ( k + 1 )} = \frac{1}{k} - \frac{1}{k + 1}$ .
The sum becomes $(1 - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + \dots + (\frac{1}{9} - \frac{1}{10})$ .
All the middle terms cancel, leaving $1 - \frac{1}{10}$ .
So the sum is $\frac{9}{10}$ .

Answer: $\frac{9}{10}$

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