First slide

Series of natural numbers

Question

Sum of the series $S = 1^{2} - 2^{2} + 3^{2} - 4^{2} + . . . - 2008^{2} + 2009^{2}$ is

Moderate

A

2019045
B

1005004
C

2000506
D

none of these

Solution

We can write $S$ as
$S = (1 - 2) (1 + 2) + (3 - 4) (3 + 1) + \dots + (2007 -$ $2008) (2007 + 2008) + 2009^{2}$
$\begin{array}{l} = - [1 + 2 + 3 + 4 + \dots + 2008] + 2009^{2} \\ = - \frac{1}{2} (2008) (2009) + 2009^{2} = 2019045 \end{array}$

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