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Question Number 189248 by 073 last updated on 13/Mar/23

Answered by mr W last updated on 13/Mar/23

$$\underset{n=1}{\sum ^{1012}}[{(2n-1)}^3-{\left(2n\right)}^3]$$$$=\underset{n=1}{\sum ^{1012}}[-12n^2+6n-1]$$$$=-12\times \frac{1012\times 1013\times (2\times 1012+1)}{6}+6\times \frac{1012\times 1013}{2}-1012$$$$=-4051\times {1012}^2$$$$\Rightarrow answer\left(B\right)$$

Commented by 073 last updated on 13/Mar/23

$$\mathrm{nice}\mathrm{solution}$$

Answered by mehdee42 last updated on 14/Mar/23

$$A=-[(1^2+1\times 2+2^2)+(3^2+3\times 4+4^2)+...+({2023}^2+2023\times 2024+{2024}^2)$$$$=-(\underset{1}{\sum ^{2024}}{i}^2+\underset{1}{\sum ^{1012}}(2n-1)\left(2n\right))$$$$=-{1012}^2\times 4051$$

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