up::[[MOC arithmétique]]
title::"$\left( \sum\limits_{k=0}^{n}a_{k} \right)^{2} = \sum\limits_{i=1}^{n}\left( \sum\limits_{j=1}^{n} a_{i}\times a_{j} \right)$"
#maths/arithmétique 

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$\left( \sum\limits_{k=0}^{n}a_{k} \right)^{2} = \sum\limits_{i=1}^{n}\left( \sum\limits_{j=1}^{n} a_{i}\times a_{j} \right)$
# Cas particuliers

 - $(a+b)^{2}= a^{2}+2ab+b^{2}$ : [[identités remarquables]]
 - $(a+b+c)^{2}= a^2+b^2+c^2+2ab+2bc+2ac$
     - ![[somme des carrés 2022-06-23 14.34.47.excalidraw|300]]
 - $(a+b+c+d)^{2}= a^{2}+b^{2}+c^{2}+d^{2}+2(ab+ac+ad+bc+bd+cd)$

# Cas général

$\left(\sum\limits_{i=0}^{n}a_{i}\right)^{2} = \sum\limits_{i=0}^{n} \left(a_{i}^{2}\right) + 2\sum\limits_{i=1}^{n-1} \left(\sum\limits_{j=i+1}^{n} a_{i}a_{j}\right) = \sum\limits_{i=1}^{n} \left(\sum\limits_{j=1}^{n} a_{i}a_{j}\right)$

description::![[carré d'une somme 2022-09-17 16.14.42.excalidraw|100%]]