Multiple choice

The sum of the series $1+\dfrac {1}{1+2}+\dfrac {1}{1+2+3}+....$ upto $10$ terms, is

  1. $\dfrac {18}{11}$
  2. $\dfrac {22}{13}$
  3. $\dfrac {20}{11}$
  4. $\dfrac {16}{9}$
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C Correct answer
Explanation

The n-th term is 1/(sum of 1 to n) = 1/(n(n+1)/2) = 2/(n(n+1)) = 2(1/n - 1/(n+1)). The sum to 10 terms is 2 * (1 - 1/2 + 1/2 - 1/3 + ... + 1/10 - 1/11) = 2 * (1 - 1/11) = 2 * (10/11) = 20/11.

AI explanation

The denominators of the series are the sums of natural numbers, so the general term is 1/[n(n + 1)/2], which simplifies to 2/[n(n + 1)]. This can be split by partial fractions into 2(1/n - 1/(n + 1)), making the overall series a telescoping sum. Evaluating this sum up to 10 terms gives 2(1 - 1/11), which equals 2(10/11) or 20/11.

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