Termenii consecutivi ai unei progresii aritmetice a, b, c au proprietatea
b=(a+c)/2
Fie r ratia progresiei
a=b-r
b=b
c=b+r
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a²=b²-2br+r²
x=a²-bc=b²-2br+r²-b²-br=r²-3br
________
b²=b²
y=b²-(b-r)*(b+r)=b²-b²+r²=r²
____________
c²=b²+2br+r²
z=c²-ab=b²+2br+r²-b²+br=r²+3br
____________
x+z=r²-3br+r²+3br=2r²=>
(x+y)/2=r²=y
deci
y=(x+z)/2 adica concluzia
