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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

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

 

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 ${100X\over p}$.