Add the previous two numbers to get the next. The ratios settle near a number that reproduces itself.
WHAT HAPPENS IF…Do successive ratios always approach the golden ratio from the same side?
Interactive exploration
Preparing your experiment…
Connect the picture to the ideaFn+1=Fn+Fn−1,φ=21+5
Dividing the recurrence by F_n gives r_n = 1 + 1/r_(n−1). If the positive ratios settle to L, then L = 1 + 1/L, whose positive solution is the golden ratio. The finite sequence plotted here illustrates this convergence.
A detail that matters
A numerical pattern illustrates convergence; it is not alone a proof of convergence.