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Golden Powers

You add 1 to the golden ratio to get its square. How do you find higher powers?

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2^n -n Numbers

Yatir from Israel wrote this article on numbers that can be written as $ 2^n-n $ where n is a positive integer.

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Poly Fibs

A sequence of polynomials starts 0, 1 and each poly is given by combining the two polys in the sequence just before it. Investigate and prove results about the roots of the polys.

Production Equation

Age 16 to 18 Challenge Level:

 

Each week a company produces $X$ units and sells $p$ per cent of its stock. How should the company plan its warehouse space? Will the stock fluctuate, or increase or decrease over time, or tend to a limit?

Initially the company has no stock. Show that, over a long period of time, the amount of stock tends to the limit ${100X\over p}$.