# A question on a summation

julypraise

## Homework Statement

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

$\sum_{i \neq j} (x_{i}^{2}y_{j}^2 - x_{i}y_{i}x_{j}y_{j}) = 2\sum_{i < j}(x_{i}^{2}y_{j}^2 + x_{j}^{2}y_{i}^2 - x_{i}y_{i}x_{j}y_{j})$

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

$\sum_{i \neq j} (x_{i}^{2}y_{j}^2 - x_{i}y_{i}x_{j}y_{j}) = \sum_{i < j}(x_{i}^{2}y_{j}^2 + x_{j}^{2}y_{i}^2 - 2x_{i}y_{i}x_{j}y_{j})$

Could you check which is right?

Thanks.

## The Attempt at a Solution

$\sum_{i \neq j} (x_{i}^{2}y_{j}^2 - x_{i}y_{i}x_{j}y_{j}) = \sum_{i < j}(x_{i}^{2}y_{j}^2 + x_{j}^{2}y_{i}^2 - 2x_{i}y_{i}x_{j}y_{j})$

$$\sum_{i \neq j} a_{ij} = \sum_{i < j} a_{ij} + \sum_{i > j} a_{ij}$$
$$\sum_{i > j} a_{ij} = \sum_{j > i} a_{ji} = \sum_{i < j} a_{ji}$$