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## Homework Statement

The Spivak's Calculus Answer Book (3ed) states that, on page 17,

[itex]\sum_{i \neq j} (x_{i}^{2}y_{j}^2 - x_{i}y_{i}x_{j}y_{j}) = 2\sum_{i < j}(x_{i}^{2}y_{j}^2 + x_{j}^{2}y_{i}^2 - x_{i}y_{i}x_{j}y_{j})[/itex]

But as I speculate, I've got the following:

[itex]\sum_{i \neq j} (x_{i}^{2}y_{j}^2 - x_{i}y_{i}x_{j}y_{j}) = \sum_{i < j}(x_{i}^{2}y_{j}^2 + x_{j}^{2}y_{i}^2 - 2x_{i}y_{i}x_{j}y_{j})[/itex]

Could you check which is right?

Thanks.

## Homework Equations

## The Attempt at a Solution

[itex]\sum_{i \neq j} (x_{i}^{2}y_{j}^2 - x_{i}y_{i}x_{j}y_{j}) = \sum_{i < j}(x_{i}^{2}y_{j}^2 + x_{j}^{2}y_{i}^2 - 2x_{i}y_{i}x_{j}y_{j})[/itex]