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For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?

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Absurdity Again

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

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Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

Generally Geometric

Age 16 to 18 Challenge Level:
Differentiate the geometric series term by term. Equate this to the derivative of the sum of the series. The required identity follows with a little simplification of the expressions to obtain terms with the required coefficients.

Now carry out this process using the formula for the infinite sum.