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I know the following equality holds from previous work

$\sum_{i=1}^{n}a_i^2\sum_{j=1}^{n}b_j^2 + \sum_{i=1}^{n}b_i^2 \sum_{j=1}^{n}a_j^2$ = $2(\sum_{i=1}^{n}a_i^2)( \sum_{i=1}^{n}b_i^2)$

But when I set values for the two sets a and b I'm getting an inequality, which makes me realize I don't quite know how to multiply these sets.

I set $a_1 = 1 a_2 = 2 a_3 = 3 b_1 = 4 b_2 =5$ and $b_3 = 6$ so I'm limiting to n=3

Now when I do it for the LHS I get 880 and the RHS I get 2156 (obviously I'm doing something wrong)

user101104
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  • This is what I'm doing: LHS: $(1^2 \times 4^2) + (2^2 \times 5 ^2) + (3^2 \times 6^2)$

    RHS: $ 2 \times (1^2 + 2^2 + 3^2) \times (4^2 + 5^2 + 6^2)$

     – user101104 Jan 28 '14 at 05:17

2 Answers2

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Why do you think you are doing something wrong? The term $a_1^2b_2^2$ is counted twice on the left and four times on the right. The right is always greater than the left by twice the sum of terms with differing subscripts.

Ross Millikan
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The indices for the sums are just dummy indices that only have meaning inside of the sum and no meaning outside of it. So using and swapping i's and j's all the time is effectively doing nothing but maybe confusing Ross Millikan.

Regarding your question, the LHS from the comment should say: ($(1^2+2^2+3^2)(4^2+5^2+6^2)+(4^2+5^2+6^2)(1^2+2^2+3^2)$ You take the product of the 2 sums and then you do it again only in different order. Your mistake was to assume that if you choose one index for the first sum it has some sort of consequence for the index of the second sum, but the second index can still be 1, 2 or 3.

Ilusion
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