The sum of the series $a - (a + d) + (a + 2 d) - (a + 3 d) + \dots$ upto $(2 n + 1)$ terms is

- n d

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a + 2 n d

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a + n d

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2 n d

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Solution

The correct option is C $a + n d$
$S = a - (a + d) + (a + 2 d) - (a + 3 d) + \dots + (a + (2 n) d)$
$\Rightarrow S = (a + (a + 2 d) + (a + 4 d) + \dots + (a + 2 n d)) - ((a + d) + (a + 3 d) + (a + 5 d) + \dots + (a + (2 n - 1) d))$
$\Rightarrow S = ((n + 1) a + (2 + 4 + \dots + 2 n) d) - n a - (1 + 3 + \dots (2 n - 1) d)$
$\Rightarrow S = a + n d$

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