What is the value of $0 - 1 + 2 - 3 + 4 - 5 + 6 - 7......... + 16 - 17 + 18 - 19 + 20$ .
$2 + 4 + 6....... + 20$ .

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Solution

$S = 0 - 1 + 2 - 3 + 4 - 5 + 6 - 7.... + 16 - 17 + 18 - 19 + 20 S = - (1 + 3 + 5 + 7 + . . . + 19) + (2 + 4 + 6 + . . . . + 20)$
To find $S$
Series $: 1 + 3 + 5 + 7 + . . . . + 19 a (f i r s t t e r m) = 1 d (c o m m o n d i f f e r e n c e) = 2 ∴ a_{n} = a + (n - 1) d 19 = 1 + (n - 1)_{2} \Rightarrow n = 10 ∴ 10 t e r m s i n s e r i e s . ∴ S_{1} = \frac{n}{2} [2 a + (n - 1) d] = \frac{10}{2} [2 + (10 - 1)_{2}] = 100$
To find $S_{2}$
Series $: 2 + 4 + 6 + . . . + 20 a = 2, d = 2 a_{n} = a + (n - 1) d \Rightarrow 20 = 2 + (n - 1)_{2} \Rightarrow n = 10 t e r m s ∴ S_{2} = \frac{n}{2} [2 a + (n - 1) d] = \frac{10}{2} [2 \times 2 + (9) 2] = 110$

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