Powers of the Golden Ratio

Some background is provided first.   Scroll down for the PROBLEMS.

Other terms used are Golden Section, Divine Proportion, Golden Mean, Divine Section, Golden Proportion, Golden Cut, Golden Number,  and others.

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Problem 1.  

Prove this without resorting to a numerical constant.   

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Problem 2.

Find the numerical value of  the Golden Ratio.

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Problem 3.

Determine the following sequence of powers and show a relationship to the Fibonacci numbers.

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Problem  4

Determine the following sequence of powers and show a relationship to the Fibonacci numbers.

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Problem 5.

If the perimeter of a circle is divided into two parts in the Golden Ratio, the the smaller central angle formed by the two arcs is called the GOLDEN ANGLE. Determine the size of the GOLDEN ANGLE, either in radians or degrees.