Multiple choice

Find the sum of the following series to n terms: $1+(1+2)+(1+2+3)+(1+2+3+4)+...$

  1. $\dfrac{n}{6}(n+1)(n+2)$
  2. $\dfrac{n}{6}(n-1)(n-2)$
  3. $\dfrac{n}{6}(n+1)(n-1)$

  4. None of these
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A Correct answer
Explanation

The n-th term of the series is the sum of the first n integers, which is n(n+1)/2. The sum of the series is the sum of n(n+1)/2 from 1 to n, which simplifies to n(n+1)(n+2)/6.

AI explanation

The k-th term of the series is the sum of the first k natural numbers, which is given by the formula k(k+1)/2. To find the sum to n terms, we evaluate the summation of k^2/2 + k/2 from k=1 to n. This results in 1/2 [n(n+1)(2n+1)/6] + 1/2 [n(n+1)/2], which simplifies to n(n+1)/4 [(2n+1)/3 + 1]. Combining the terms inside the brackets gives (2n+4)/3, making the entire expression n(n+1)(n+2)/6.

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