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Hello, I am a recently graduated high school senior going off to college soon, and over the summer I have spend some of my free time experimenting with different problems in mathematics. This is one that I have spent many days on, and have come to a dead end.

Like the title says, I am attempting to generalize the squaring of any sum of variables

[tex] (x_1+ \dots + x_n)^2 [/tex]

So far I've just experimented with multiplying it out with different values of n and trying to spot a pattern.

For n = 3 it comes out to be

[tex] x^2_1 +x^2_2 +x^2_3 + 2(x_1 x_2 + x_1 x_3 + x_2 x_3) [/tex]

The pattern seems to be the squares of the variables + 2 times all the combinations of 2 variables multiplied together.

I thought higher values of n would help me get a general pattern so I chose n = 6

[tex] (x_1 + \dots + x_6)^2 = x^2_1 + x^2_2 + x^2_3 + x^2_4 + x^2_5 + x^2_6 + 2(x_1 x_2 + x_1 x_3 + x_1 x_4 + x_1 x_5 + x_1 x_6 + x_2 x_3 + x_2 x_4 + x_2 x_5 + x_2 x_6 + x_3 x_4 + x_3 x_5 + x_3 x_6 + x_4 x_5 + x_4 x_6 + x_5 x_6) [/tex]

Since there are so many terms to be added I thought using Sigma notation would help. The same formula above in Sigma notation is

[tex] ( \sum_{i=1}^6 x_i )^2 = \sum_{i=1}^6 x^2_i + 2( \sum_{i=2}^6 x_1 x_i + \sum_{i=3}^6 x_2 x_i +\sum_{i=4}^6 x_3 x_i + \sum_{i=5}^6 x_4 x_i + x_5 x_6 ) [/tex]

This is obviously not a very compact formula, and I was wondering if there is a way to combine the sums of different indices. Or, if there is a better way to go about this.

Like the title says, I am attempting to generalize the squaring of any sum of variables

[tex] (x_1+ \dots + x_n)^2 [/tex]

So far I've just experimented with multiplying it out with different values of n and trying to spot a pattern.

For n = 3 it comes out to be

[tex] x^2_1 +x^2_2 +x^2_3 + 2(x_1 x_2 + x_1 x_3 + x_2 x_3) [/tex]

The pattern seems to be the squares of the variables + 2 times all the combinations of 2 variables multiplied together.

I thought higher values of n would help me get a general pattern so I chose n = 6

[tex] (x_1 + \dots + x_6)^2 = x^2_1 + x^2_2 + x^2_3 + x^2_4 + x^2_5 + x^2_6 + 2(x_1 x_2 + x_1 x_3 + x_1 x_4 + x_1 x_5 + x_1 x_6 + x_2 x_3 + x_2 x_4 + x_2 x_5 + x_2 x_6 + x_3 x_4 + x_3 x_5 + x_3 x_6 + x_4 x_5 + x_4 x_6 + x_5 x_6) [/tex]

Since there are so many terms to be added I thought using Sigma notation would help. The same formula above in Sigma notation is

[tex] ( \sum_{i=1}^6 x_i )^2 = \sum_{i=1}^6 x^2_i + 2( \sum_{i=2}^6 x_1 x_i + \sum_{i=3}^6 x_2 x_i +\sum_{i=4}^6 x_3 x_i + \sum_{i=5}^6 x_4 x_i + x_5 x_6 ) [/tex]

This is obviously not a very compact formula, and I was wondering if there is a way to combine the sums of different indices. Or, if there is a better way to go about this.

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