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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
Taking $X_n$ as the amount of stock at the end of week $n$, you need to solve the difference equation (recurrence relation) $$X_{n+1} = X + (1 - {p\over 100})X_n$$ Put $X_n = Y_n - C$ then choose $C$ such that $$Y_{n+1} = (1 - {p\over 100})Y_n$$ and consider the values of this expression for $Y_n, Y_{n-1}, Y_{n-2}, ... Y_1, Y_0$.