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# Log Attack

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### Infinite Continued Fractions

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Age 16 to 18

Challenge Level

- Problem
- Student Solutions

Solve the equation $a^x + b^x = 1$ where $0< a, b < 1$ and $a
+ b < 1$, in the special cases:

(i) $a = b\quad $ (ii) $a = {1\over 2}, \ b={1\over 4}\quad $

You can find exact solutions to the equation $a^x + b^x = 1$ in special cases like (i) and (ii).

(i) $a = b\quad $ (ii) $a = {1\over 2}, \ b={1\over 4}\quad $

You can find exact solutions to the equation $a^x + b^x = 1$ in special cases like (i) and (ii).

More generally you will need to use a numerical method for
finding approximate solutions. See
Equation Attack.

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

In this article we are going to look at infinite continued fractions - continued fractions that do not terminate.

Explore the hyperbolic functions sinh and cosh using what you know about the exponential function.