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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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And So on - and on -and On

Can you find the value of this function involving algebraic fractions for x=2000?

Poly Fibs

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Use the defining relation for these polynomials to write $P_6$ in terms of $P_5$ and $P_4$ then in terms of $P_3$ and $P_2$ and the result follows.

The results for $P_8$ and for $P_{10}$ are proved similarly.