Ciag (a,b,c) jest geometryczny.Wykaz, ze (a+b+c)(a-b+c)=a^2+ b^2+c^2.... Question from @KikixOx

December 2018 1 8 Report

Ciag (a,b,c) jest geometryczny.Wykaz, ze (a+b+c)(a-b+c)=a^2+ b^2+c^2.

Paawełek Jest to ciąg geometryczny o jakimś ilorazie q, tak więc:
$b=aq \\ c=aq^2$

Podstawiając:

$(a+b+c)(a-b+c)=(a+aq+aq^2)(a-aq+aq^2)= \\ \\ =a(q^2+q+1) \cdot a(q^2-q+1) = \\ \\ =a^2(q^2+q+1)(q^2-q+1)= \\ \\ =a^2(q^4-q^3+q^2+q^3-q^2+q+q^2-q+1)= \\ \\ =a^2(q^4+q^2+1)= \\ \\ =a^2+a^2q^2+a^2q^4= \\ \\=a^2+(\underbrace{aq}_{b})^2 + (\underbrace{aq^2}_{c})^2 = \\ \\ = a^2+b^2+c^2$

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