The sum of the first 20 terms of the series $\frac{1}{5 \times 6} + \frac{1}{6 \times 7} + \frac{1}{7 \times 8} + \dots$ is

### Concept & Formula This is a telescoping series where each individual term can be split using partial fractions: $$ \frac{1}{n(n+1)} = \frac{1}{n} - \frac{1}{n+1} $$ ### Step-by-Step Solution * Identify the form of the $n$-th term: $$ T_n = \frac{1}{(n+4)(n+5)} $$ * Determine the 20th term of this series: For $n = 1$, the term is $\frac{1}{5 \times 6}$. For $n = 20$, the term is $\frac{1}{(20+4) \times (20+5)} = \frac{1}{24 \times 25}$. * Expand the sum of the $20$ terms using the splitting method: $$ S_{20} = \left(\frac{1}{5} - \frac{1}{6}\right) + \left(\frac{1}{6} - \frac{1}{7}\right) + \dots + \left(\frac{1}{24} - \frac{1}{25}\right) $$ * Cancel out the intermediate terms (telescoping property): $$ S_{20} = \frac{1}{5} - \frac{1}{25} $$ * Compute the final fractional and decimal value: $$ S_{20} = \frac{5 - 1}{25} = \frac{4}{25} $$ $$ \frac{4}{25} = 0.16 $$ ### Exam Strategy & Shortcut For any standard telescoping series with a difference of $1$ in the denominator factors, the sum is always: $$ \text{Sum} = \text{First Factor}^{-1} - \text{Last Factor}^{-1} $$ Here, the first factor is $5$, and the last factor after $20$ terms is $5 + 20 = 25$. $$ \frac{1}{5} - \frac{1}{25} = \frac{4}{25} = 0.16 $$ ### Common Pitfall Mistakenly calculating the 20th term's denominator as $20 \times 21$ instead of tracking the initial offset starting at $5$. Always verify the first term match before computing the final term. ### Final Answer **Therefore, the correct answer is $0.16$.**