Question

Tìm x ϵ N biết \(\dfrac{1}{4}.\dfrac{2}{6}.\dfrac{3}{8}.\dfrac{4}{10}......\dfrac{14}{30}.\dfrac{15}{32}=\dfrac{1}{2^x}\)

Answer 1

\(\dfrac{1}{4}\cdot\dfrac{2}{6}\cdot\dfrac{3}{8}\cdot\dfrac{4}{10}\cdot...\cdot\dfrac{14}{30}.\dfrac{15}{32}=\dfrac{1}{2^x}\)

\(\Rightarrow\dfrac{1\cdot2\cdot3\cdot4\cdot...\cdot14\cdot15}{4\cdot6\cdot8\cdot10\cdot...\cdot30\cdot32}=\dfrac{1}{2^x}\)

\(\Rightarrow\dfrac{1\cdot2\cdot3\cdot4\cdot...\cdot14\cdot15}{2\cdot4\cdot6\cdot8\cdot10\cdot...\cdot30\cdot32}=\dfrac{1}{2^{x+1}}\)

\(\Rightarrow\dfrac{1}{2^{15}\cdot32}=\dfrac{1}{2^{x+1}}\)

\(\Rightarrow2^{15}.2^5=2^{x+1}\)

\(\Rightarrow2^{20}=2^{x+1}\)

\(\Rightarrow x+1=20\Rightarrow x=19\)

Vậy x = 19.