The Golden Ratio

The golden ratio is found from the limit, as n tends to infinity, of the ratio of two successive terms of the Fibonacci sequence: x(n)=x(n-1)+x(n-2).

Divide the recurrence relation by x(n-1) and set r(n)=x(n)/x(n-1) to show that the limit of the ratio, r, satisfies r=1+1/r.

Solve this equation using the quadratic formula, using Newton-Raphson and using another fixed point method.

---

#sequences #equationsolving #numericalmethods #goldenratio

---