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Converging Product

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?

Rain or Shine

Predict future weather using the probability that tomorrow is wet given today is wet and the probability that tomorrow is wet given that today is dry.

There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Golden Eggs

Age 16 to 18
Challenge Level

ellipse and annulus
1) An ellipse with semi axes $a$ and $b$ fits between two circles of radii $a$ and $b$ (where $b> a$) as shown in the diagram. If the area of the ellipse is equal to the area of the annulus what is the ratio $b:a$?

(2) Find the value of $R$ if this sequence of 'nested square roots' continues indefinitely: $$R=\sqrt{1 + \sqrt{1 + \sqrt {1 + \sqrt {1 + ...}}}}.$$