First slide

Series of natural numbers

Question

Sum to n terms of the series $1 + (1 + 2) + (1 + 2 + 3) + \dots$ is

Moderate

A

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

$\frac{1}{3} n (n + 1) (n + 2)$
C

$\frac{1}{4} n (n + 1) (2 n + 1)$
D

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

Solution

$\begin{array}{l} t_{r} = 1 + 2 + \dots + r = \frac{1}{2} r (r + 1) = \frac{1}{2} (r^{2} + r) \\ \sum_{r = 1}^{n} t_{r} = \frac{1}{2} [\frac{1}{6} n (n + 1) (2 n + 1) + \frac{1}{2} n (n + 1)] \\ = \frac{1}{6} n (n + 1) (n + 2) \end{array}$

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