What is the value of ( $2^{2} + 4^{2} + 6^{2} + . . . + 20^{2}$ )-( $1^{2} + 3^{2} + 5^{2} + . . . + 19^{2}$ )?

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Solution

$(2^{2} + 4^{2} + . . . . + 20^{2}) - (1^{2} + 3^{2} + . . . + 19^{2}) c a n b e b r o k e n a s, (2^{2} - 1^{2}) + . . . . + (20^{2} - 19^{2}), u s i n g (a^{2} - b^{2}) = (a + b) (a - b), ⟹ (2 + 1) (1) + (3 + 4) (1) + . . . . + (20 + 19) (1), ⟹ 3 + 7 + . . . .39, a r i t h m e t i c p r o g r e s s i o n w i t h 10 t e r m s, ⟹ \frac{10}{2} (3 + 39) = 210$

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