Golden ratio

When is a Fibonacci sequence also a geometric sequence? When the ratio of successive terms is the golden ratio!

Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.

Nick Lord says "This problem encapsulates for me the best features of the NRICH collection."

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

Solve an equation involving the Golden Ratio phi where the unknown occurs as a power of phi.