Square of a generic sum of elements

The formula for computing the square of a generic sum of terms is as follows:

\left(\sum_{k=1}^{K}a_{k}\right)^{2}=\sum_{k=1}^{K}a_{k}^{2}+2\sum_{j=1}^{K}% \sum_{i<j}a_{i}a_{j}

We can prove this property by induction, considering that it holds for K=2, since

(a+b)^{2}=a^{2}+b^{2}+2ab

and that if the property holds for a generic K, then it holds also for K+1, as is proven in the following passages:

\aligned\left(\sum_{k=1}^{K+1}a_{k}\right)^{2}&=\left(\sum_{k=1}^{K}a_{k}+a_{K% +1}\right)^{2}=\left(\sum_{k=1}^{K+1}a_{k}\right)^{2}+a_{K+1}^{2}+2\sum_{k=1}^% {K}a_{k}a_{K+1}\\ &=\sum_{k=1}^{K}a_{k}^{2}+2\sum_{j=1}^{K}\sum_{i<j}a_{i}a_{j}+a_{K+1}^{2}+2% \sum_{k=1}^{K}a_{k}a_{K+1}=\\ &=\sum_{k=1}^{K+1}a_{k}^{2}+2\sum_{j=1}^{K+1}\sum_{i<j}a_{i}a_{j}

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Title	Square of a generic sum of elements
Canonical name	SquareOfAGenericSumOfElements
Date of creation	2013-03-22 19:34:48
Last modified on	2013-03-22 19:34:48
Owner	mat (27197)
Last modified by	mat (27197)
Numerical id	8
Author	mat (27197)
Entry type	Definition
Classification	msc 15-01