Question

Aflatimedia aritmetica si geometrica a numerelor a=√√5+1:√5-1; b=√√5-1:√5+1;

Answer 1

a=√(√5+1):(√5-1)=√(√5+1)(√5+1)/(√5-1)(√5+1)=√(5+1+2√5)/(5-1)=
=√(6+2√5)/4=1/2 * √(6+2√5)= 1/2*√(6+√20)=1/2*(√5+1)=√5/2+1/2
√(6+√20)=
=√(6+√(36-20)/2+√(6-√(36-20)/2=
=√(6+√16)/2+√(6-√16)/2=
=√(6+4)/2+√(6-4)/2=
=√10/2+√2/2=
=√5+1
b=√(√5-1):(√5+1)=√(√5-1)(√5-1)/(√5+1)(√5-1)=√(5+1-2√5)/(5-1)=
=√(6-2√5)/4=1/2 * √(6-2√5)=1/2 * √(6-√20)= 1/2*(√5-1)=√5/2-1/2
√(6-√20)=
=√(6+√(36-20)/2-√(6-√(36-20)/2=
=√(6+√16)/2-√(6-√16)/2=
=√(6+4)/2-√(6-4)/2=
=√10/2-√2/2=
=√5-1
ma=(a+b)/2=(√5/2+1/2+√5/2-1/2)/2=√5/2
mg=√a=√(√5/2+1/2)(√5/2-1/2)=√(5/4-1/4)=√4/4=√1=1