Show that the arithmetic mean, geometric mean and harmonic mean of
a and b can be the lengths of the sides of a right-angles triangle
if and only if a = bx^3, where x is the Golden Ratio.

Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.

Golden Powers

Age 16 to 18 Challenge Level:

The famous golden ratio is $g={\sqrt5 + 1 \over 2}$. Prove that $g^2=g+1$.

Let $g^n = a_ng + b_n$. Find the sequences of coefficients $a_n$ and $b_n$.