Skoro liczby a, b, c tworzą ciąg geometryczny to:

  $b=aq,c=a{q}^2$  

$a)L=\left(a+b+c\right)\left(a-b+c\right)=\left(a+aq+a{q}^2\right)\left(a-aq+a{q}^2\right)=\left(a+a{q}^2+aq\right)\left(a+a{q}^2-aq\right)={\left(a+a{q}^2\right)}^2-{\left(aq\right)}^2=a^2+2a\cdot a{q}^2+a^2{q}^4-a^2{q}^2=a^2\left(1+2q^2+q^4-q^2\right)=a^2\left(1+q^2+q^4\right)=a^2+a^2{q}^2+a^2{q}^4=a^2+{\left(aq\right)}^2+{\left(a{q}^2\right)}^2=a^2+b^2+c^2=P$  

 

$b)L=a^2{b}^2{c}^2\left(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}\right)=\frac{b^2{c}^2}{a}+\frac{a^2{c}^2}{b}+\frac{a^2{b}^2}{c}=\frac{{\left(aq\right)}^2\cdot {\left(a{q}^2\right)}^2}{a}+\frac{a^2\cdot {\left(a{q}^2\right)}^2}{aq}+\frac{a^2\cdot {\left(aq\right)}^2}{a{q}^2}=\frac{a^2{q}^2\cdot a^2{q}^4}{a}+\frac{a^2\cdot a^2\cdot q^4}{aq}+\frac{a^2\cdot a^2\cdot q^2}{a{q}^2}=a^3{q}^6+a^3{q}^3+a^3=a^3+{\left(aq\right)}^3+{\left(a{q}^2\right)}^3=a^3+b^3+c^3=P$