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# Golden Eggs

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: $$R=\sqrt{1 + \sqrt{1 + \sqrt {1 + \sqrt {1 + ...}}}}.$$

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Age 16 to 18

Challenge Level

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: $$R=\sqrt{1 + \sqrt{1 + \sqrt {1 + \sqrt {1 + ...}}}}.$$

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)... ?

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.

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?