Binomial theorem proof in darija, Essays (high school) of Mathematics

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Démonstration de binôme de Newton en Darija

May 22, 2024

MRX

I -Lmochkil:

Fhad l’article bghit ndemontrer lformule lmechhora lme3rofa bbinôme de Newton li hiya hadi :

(x + y)n^ =

∑^ n

k=

Cnk xkyn−k

(Machi b Récurrence btabi3t l7al )

II -Lborhan:

Bach nbyno had l3ala9a aji nketboha btari9a akhra. Pour n ∈ N:

(x + y)n^ = (x + y)(x + y)... (x + y) ︸ ︷︷ ︸ bzzaf dlmrat (n)

Hna ghanla7do wahd lhaja hiya fach anjiw nnechro had lkhir anakhdo wahd l’élément (x ola y li bghina) mn kola (x + y) o anderboh f wa7ed l’élément (x ola y tani) mn hadak li b3do.

Ye3ni fwahd l7ala anakhdo xn^ o f7ala akhra anakhdo xn−^1 × y oghada 7ta kanweslo l yn.Donc bach n3ememo hadchi le terme générale aykon howa ”xkyn−k” (ola ”xn−kyk” li 7bat khatrek) btab3t l7al k ∈ { 0 , 1 ,... , n}.

btabi3t l7al maghanakhdohomch mera w7da kola terme anakhdoh wahd l3adad dlmerrat.

YE3niiii l3ala9a atwli bhal heka:

(x + y)n^ = anxn^ + an− 1 xn−^1 y + an− 2 xn−^2 y^2 + · · · + a 0 yn

Db khesna njebdo had l’a3dad {a 0 ,... , an}!!

Bach njebdo had les coefficients aykhesna n3tabro had les termes li homa kola (x + y) des ensembles li fihom gha joj dl3anasir li homa x et y.ye3ni (x + y) ayweli {x, y}.

O 3endna 3awtani le terme ”akxkyn−k” kay3ni anaho khdina x wahd l3adad dlmerrat li howa k mn l’ensemble {x, y} ol3adad dlmerrat li khdina biha y howa lba9i li howa n − k.Donc hna an- dekhlo wahd la notion mohima li hiya le coefficient binomial (ola la combinaison kima 3refnha flproba hh) ”Cnk ”(ola

(n k

) li kat3etina l3adad dyal l’i7timalat lmomkina dyal nakhdo wa7d les ensembles li l3adad dl3anasir fihom howa k mn wa7d l’ensemble fih wahd l3adad dl3anasir n.

Ye3nii le coefficient ak aykon howa had l3adad dyal l’i7timalat li howa Ckn.

Aloors l3ala9a atwli b7al haka :

(x + y)n^ = Cnn xn^ + C nn −^1 xn−^1 y + C nn −^2 xn−^2 y^2 + · · · + C n^0 yn

O hna ankono demontrina l3ala9a li bghina li hiya:

(x + y)n^ =

∑^ n

k=

Cnk xkyn−k

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