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The following equation can be easily proved with induction (using $(a+b)^2=a^2+2ab+b^2$):

$$\left(\sum\limits_{i=1}^n a_i\right)^2 = \sum\limits_{i=1}^n a_i^2 + 2\sum\limits_{1\leq i<j\leq n} a_ia_j$$

Do you have some intuitive explanation why is this equation true?

Elias Costa
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zvisofer
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    The simplest explanation I can think of is to draw up a square with sides $\sum a_i$ long and subdivided for each $i$. Then you can just see what the big square is built up from (I don't have the means to produce such an image right now). Your identity is two different ways of calculating the area of the square (LHS is just $\text{side}^2$, RHS is the sum of all the small squares and rectangles) – Arthur Oct 28 '13 at 13:00

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Added an illustration of Arthur's answer:

"[..] draw up a square with sides ∑ai long and subdivided for each i. Then you can just see what the big square is built up from. Your identity is two different ways of calculating the area of the square (LHS is just side^2, RHS is the sum of all the small squares and rectangles)"

squares

zvisofer
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Imagine multiplying a number with 10 digits with itself, grade-school style. Then the product is the sum of the squares per digit, and twice of every combination product of two different digits.

aaa
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