(1) The areas of the ellipse and the annulus are pab and p(b² - a²) and so

b² - a²

= ab

(b/a)² - (b/a) - 1

= 0

and so, as b/a is positive, b/a = (1 + Ö5)/2 which is the Golden Ratio.

(2)As

R= æ
ú
Ö

1 + æ
Ö

1 +
Ö

1 + [(1 + ...)]

is an infinite 'nested sequence' we have

R² - 1 = R

so we find the value of R by solving the equation R² - R - 1 = 0 and, as R must be positive, R=(1 + Ö5)/2, the Golden Ratio.