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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?

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Not Continued Fractions

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

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Fair Shares?

A mother wants to share a sum of money by giving each of her children in turn a lump sum plus a fraction of the remainder. How can she do this in order to share the money out equally?

Largest Expression

Stage: 4 Short Challenge Level: Challenge Level:2 Challenge Level:2

Given that $x$ is in $(0,1)$, we may deduce that $x^2+x> x^2> x^3> x^4$ and also that $x^2+x> x^2+x^3 = x(x+x^2)$.

This problem is taken from the UKMT Mathematical Challenges.
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