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# Archimedes Numerical Roots

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions

A sequence of numbers $x_1, x_2, x_3, ... ,$ starts with $x_1 = 2$,
and, if you know any term $x_n$, you can find the next term
$x_{n+1}$ using the formula $x_{n+1} = \frac{1}{2} \left( x_n +
\frac{3}{x_n} \right)$.

Find the value of sqrt(2+sqrt3)-sqrt(2-sqrt3)and then of cuberoot(2+sqrt5)+cuberoot(2-sqrt5).

Find the smallest numbers a, b, and c such that: a^2 = 2b^3 = 3c^5 What can you say about other solutions to this problem?