2025 AMC 12B Problem 7

Problem 7 of 25EasierAlgebra

What is the value of

\sum_{n=2}^{255} \frac{\log_2\left(1 + \frac{1}{n}\right)}{(\log_2 n)(\log_2(n+1))}?

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Solution

Let

a_n = \log_2 n.

The numerator equals

a_{n+1} - a_n,

so each term is

\dfrac{a_{n+1} - a_n}{a_n a_{n+1}} = \dfrac{1}{a_n} - \dfrac{1}{a_{n+1}}.

Telescoping from

n = 2

255

leaves

\dfrac{1}{\log_2 2} - \dfrac{1}{\log_2 256} = 1 - \dfrac{1}{8}

= \dfrac{7}{8}.

Thus, the correct answer is C.