Question

Write the sum of 20 terms of the series \[1 + \frac{1}{2}(1 + 2) + \frac{1}{3}(1 + 2 + 3) + . . . .\]

Answer 1

Let the nth term be \[a_n\]

Here, 

\[a_n = \frac{1}{n}\left( 1 + 2 + 3 + . . . + n \right) = \left( \frac{n + 1}{2} \right)\]

We know:

\[S_n = \sum^n_{k = 1} a_k\]

Thus, we have:

\[S_{20} = \sum^{20}_{k = 1} a_k\]

\[= \frac{1}{2}\left[ \sum^{20}_{k = 1} \left( k + 1 \right) \right]\]

\[ = \frac{1}{2}\left[ \sum^{20}_{k = 1} k + 20 \right]\]

\[ = \frac{1}{2}\left[ \frac{20\left( 21 \right)}{2} + 20 \right]\]

\[ = \frac{1}{2}\left[ 230 \right]\]

\[ = 115\]