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← AMC 12B 2025← Problems

AMC 12B 2025 (Problem 7)

Sequences & SeriesEasy
What is the value of ∑n=2255log⁡2(1+1n)(log⁡2n)(log⁡2(n+1))?\sum_{n=2}^{255}\frac{\log_2\left(1+\frac1n\right)}{(\log_2 n)(\log_2(n+1))}?
(A)  34\text{(A)}\;\frac34(B)  1−1log⁡2255\text{(B)}\;1-\frac{1}{\log_2 255}(C)  78\text{(C)}\;\frac78(D)  1516\text{(D)}\;\frac{15}{16}(E)  1\text{(E)}\;1