Suppose that 4^x_1 = 5, 5^x_2 = 6, 6^x_3 = 7, …, 127^x_124 = 128. What is x_1 x_2 … x_124?

2005 AMC 12B Problem 13

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13.

Suppose that

4^{x_1} = 5,

5^{x_2} = 6,

6^{x_3} = 7,

\ldots,

127^{x_{124}} = 128.

What is

x_1 x_2 \cdots x_{124}?

2

\dfrac{5}{2}

3

\dfrac{7}{2}

4

Each equation gives

x_k = \log_{k+3}(k+4)

The product telescopes to

\log_4 128

From

4^{x_1} = 5

we get

x_1 = \log_4 5,

and in general

x_k = \log_{k+3}(k+4).

The product telescopes:

\begin{aligned} &x_1 x_2 \cdots x_{124} \\ &= \log_4 5 \cdot \log_5 6 \cdots \log_{127} 128 \\ &= \log_4 128. \end{aligned}

Since

128 = 2^7

and

4 = 2^2,

this equals

\dfrac{7\log 2}{2\log 2} = \dfrac72.

Thus, the correct answer is D.