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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:
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 $Y_n, Y_{n-1}, Y_{n-2}, ... Y_1, Y_0$.