# How to properly expand this summation?

I have the following summation:

And I'm tying to expand it... I'm not sure this is correct and I need help fixing it.

• i = {0, ..., 9}
• j = {1, ..., 8}

Here's my attempt:

x01 + x02 + x03 + x04 + x05 + x06 + x07 + x08 + x12 + x13 + x14 + x15 + x16 + x17 + x18 + x23 + x24 + x25 + x26 + x27 + x28 + x34 + x35 + x36 + x37 + x38 + x45 + x46 + x47 + x48 + x56 + x57 + x58 + x67 + x68 + x78 + x10 + x20 + x30 + x40 + x50 + x60 + x70 + x80 + x21 + x31 + x41 + x51 + x61 + x71 + x81 + x32 + x42 + x52 + x62 + x72 + x82 + x43 + x53 + x63 + x73 + x83 + x54 + x64 + x74 + x84 + x65 + x75 + x85 + x76 + x86 + x87 = 2v11;

x01 + x02 + x03 + x04 + x05 + x06 + x07 + x08 + x12 + x13 + x14 + x15 + x16 + x17 + x18 + x23 + x24 + x25 + x26 + x27 + x28 + x34 + x35 + x36 + x37 + x38 + x45 + x46 + x47 + x48 + x56 + x57 + x58 + x67 + x68 + x78 + x10 + x20 + x30 + x40 + x50 + x60 + x70 + x80 + x21 + x31 + x41 + x51 + x61 + x71 + x81 + x32 + x42 + x52 + x62 + x72 + x82 + x43 + x53 + x63 + x73 + x83 + x54 + x64 + x74 + x84 + x65 + x75 + x85 + x76 + x86 + x87 = 2v12;

x01 + x02 + x03 + x04 + x05 + x06 + x07 + x08 + x12 + x13 + x14 + x15 + x16 + x17 + x18 + x23 + x24 + x25 + x26 + x27 + x28 + x34 + x35 + x36 + x37 + x38 + x45 + x46 + x47 + x48 + x56 + x57 + x58 + x67 + x68 + x78 + x10 + x20 + x30 + x40 + x50 + x60 + x70 + x80 + x21 + x31 + x41 + x51 + x61 + x71 + x81 + x32 + x42 + x52 + x62 + x72 + x82 + x43 + x53 + x63 + x73 + x83 + x54 + x64 + x74 + x84 + x65 + x75 + x85 + x76 + x86 + x87 = 2v13;

x01 + x02 + x03 + x04 + x05 + x06 + x07 + x08 + x12 + x13 + x14 + x15 + x16 + x17 + x18 + x23 + x24 + x25 + x26 + x27 + x28 + x34 + x35 + x36 + x37 + x38 + x45 + x46 + x47 + x48 + x56 + x57 + x58 + x67 + x68 + x78 + x10 + x20 + x30 + x40 + x50 + x60 + x70 + x80 + x21 + x31 + x41 + x51 + x61 + x71 + x81 + x32 + x42 + x52 + x62 + x72 + x82 + x43 + x53 + x63 + x73 + x83 + x54 + x64 + x74 + x84 + x65 + x75 + x85 + x76 + x86 + x87 = 2v14;

x01 + x02 + x03 + x04 + x05 + x06 + x07 + x08 + x12 + x13 + x14 + x15 + x16 + x17 + x18 + x23 + x24 + x25 + x26 + x27 + x28 + x34 + x35 + x36 + x37 + x38 + x45 + x46 + x47 + x48 + x56 + x57 + x58 + x67 + x68 + x78 + x10 + x20 + x30 + x40 + x50 + x60 + x70 + x80 + x21 + x31 + x41 + x51 + x61 + x71 + x81 + x32 + x42 + x52 + x62 + x72 + x82 + x43 + x53 + x63 + x73 + x83 + x54 + x64 + x74 + x84 + x65 + x75 + x85 + x76 + x86 + x87 = 2v15;

x01 + x02 + x03 + x04 + x05 + x06 + x07 + x08 + x12 + x13 + x14 + x15 + x16 + x17 + x18 + x23 + x24 + x25 + x26 + x27 + x28 + x34 + x35 + x36 + x37 + x38 + x45 + x46 + x47 + x48 + x56 + x57 + x58 + x67 + x68 + x78 + x10 + x20 + x30 + x40 + x50 + x60 + x70 + x80 + x21 + x31 + x41 + x51 + x61 + x71 + x81 + x32 + x42 + x52 + x62 + x72 + x82 + x43 + x53 + x63 + x73 + x83 + x54 + x64 + x74 + x84 + x65 + x75 + x85 + x76 + x86 + x87 = 2v16;

x01 + x02 + x03 + x04 + x05 + x06 + x07 + x08 + x12 + x13 + x14 + x15 + x16 + x17 + x18 + x23 + x24 + x25 + x26 + x27 + x28 + x34 + x35 + x36 + x37 + x38 + x45 + x46 + x47 + x48 + x56 + x57 + x58 + x67 + x68 + x78 + x10 + x20 + x30 + x40 + x50 + x60 + x70 + x80 + x21 + x31 + x41 + x51 + x61 + x71 + x81 + x32 + x42 + x52 + x62 + x72 + x82 + x43 + x53 + x63 + x73 + x83 + x54 + x64 + x74 + x84 + x65 + x75 + x85 + x76 + x86 + x87 = 2v17;

x01 + x02 + x03 + x04 + x05 + x06 + x07 + x08 + x12 + x13 + x14 + x15 + x16 + x17 + x18 + x23 + x24 + x25 + x26 + x27 + x28 + x34 + x35 + x36 + x37 + x38 + x45 + x46 + x47 + x48 + x56 + x57 + x58 + x67 + x68 + x78 + x10 + x20 + x30 + x40 + x50 + x60 + x70 + x80 + x21 + x31 + x41 + x51 + x61 + x71 + x81 + x32 + x42 + x52 + x62 + x72 + x82 + x43 + x53 + x63 + x73 + x83 + x54 + x64 + x74 + x84 + x65 + x75 + x85 + x76 + x86 + x87 = 2v18;

In case you find it helpful and know how to code, the following output was generated with the following PHP snippet:

for($k = 1;$k <= 8; $k++) {$output = null;

for($i = 0;$i <= 9; $i++) { for($j = 1; $j <= 8;$j++) {
if($i <$j) $output .= "x$i$j + "; } } for($i = 0; $i <= 9;$i++) {
for($j = 1;$j <= 8; $j++) { if($j > $i)$output .= "x$j$i + ";
}
}

$output = substr($output, 0, strlen($output) - 3);$output .= " = 2v1$k;"; echo "$output<br><br>";
}

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The problem is, in your expansion, you are not realizing that $j$ is a fixed value. $j$ shouldn't vary in your sums on the left side, so, for example, when $j=3$, you want: $\sum_{i<3} x_{i3} + \sum_{i>3} x_{3i} = 2v_{13}$ – Thomas Andrews Jan 4 '12 at 20:30
So, you're saying I should have a single expression instead of the 8 above? Because j goes from 1 to 8... – Ricardo Amaral Jan 4 '12 at 20:35
No, for each of the eight values of $j$, the sum should be only of $9$ elements. – Thomas Andrews Jan 4 '12 at 20:38
I think I got it. Makes sense now :) If you could move the comments above to an answer below, I would gladly accept it as the correct answer. – Ricardo Amaral Jan 4 '12 at 21:32

The problem is that, for each $j$, the sums do not vary over $j$.
For example, for $j=3$, the expansion would be:
$$2v_{13} = x_{03} + x_{13} + x_{23} + x_{34} + x_{35} + x_{36} + x_{37} + x_{38} + x_{39}$$
Part of the problem is that $j$ is so often used as an index in a sigma ($\sum$) expression that it made you think it should vary. If you replaced all of the $j$ with, say, $N$, it would seem more obvious, I think.
What really confused me was not the index itself, more like I assumed the summation was for i and j when it was actually only for i. Thank you. – Ricardo Amaral Jan 4 '12 at 22:29