First slide

Sigma operation series

Question

$\sum_{i = 1}^{n} \sum_{j = 1}^{i} \sum_{k = 1}^{j} 1 =$

Moderate

A

$\frac{n (n + 1) (2 n + 1)}{6}$
B

$\frac{n (n + 1)}{2}$
C

${(\frac{n (n + 1)}{2})}^{2}$
D

$\frac{n (n + 1) (n + 2)}{6}$

Solution

We have
$\sum_{i = 1}^{n} \sum_{j = 1}^{i} \sum_{k = 1}^{j} 1 = \sum_{i = 1}^{n} \sum_{j = 1}^{i} j$
$\begin{array}{l} = \sum_{i = 1}^{n} \frac{i (i + 1)}{2} = \frac{1}{2} [\sum_{i = 1}^{n} i^{2} + \sum_{i = 1}^{n} i] \\ - \frac{1}{2} [\frac{n (n + 1) (2 n + 1)}{6} + \frac{n (n + 1)}{2}] \\ = \frac{n (n + 1) (n + 2)}{6} \end{array}$

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