Another Radical Expression

This problem was submitted by Sasha Joseph, this year's high scoring sophomore at the MAML state math meet.

If f(x) = 3b²^x, with b a constant greater than zero, find the value of the following expression in terms of b.

f(1)-

f(2)+

f(3)-

f(4)

...

View the solution

Another Radical Expression Solution

In order to solve this problem, first we recognize that the b's can be factored out of the entire expression. This is because the exponent of b is continually doubling within the nested radicals.

The result is:

b*sqrt(3 - sqrt(3 + sqrt(3 - ...)...)

Some solvers stopped here; however, this is not completely simplified.

Let X = sqrt(3 - sqrt(3 + sqrt(3 - ...)...)
X = sqrt(3 - sqrt(3 + X))

X2 = 3 - sqrt(3 + X)

X2 - 3 = sqrt(3 + X)

X4 - 6X2 + 9 = 3 + X

X4 - 6X2 - X + 6 = 0

Factoring this gives:

(X-1)(X+2)(X2-x-3) = 0

By testing the various roots, we can see that X=1 is the correct value.

Thus, the answer is 1b = b

Graeme's Number

Arithmetic and Geometric

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