Sequences & Series
Product of f(N) — integer part of reciprocal
MJAT_TS6_P2
Grade 12
Question:
Let $f(N)=1-\dfrac{1}{\log_2(N+1)}$. The integral part of $\dfrac{1}{f(31)\cdot f(63)\cdot f(127)\cdot f(255)\cdots}$ (10 terms) equals:
Step-by-Step Solution
Key Concept: $f(2^k-1)=1-1/k=(k-1)/k$. Terms: $f(31)=4/5$, $f(63)=5/6$, ..., $f(2^{14}-1)=13/14$. Product $=\frac{4}{5}\cdot\frac{5}{6}\cdots\frac{13}{14}=\frac{4}{14}=\frac{2}{7}$.
Integral part $=\mathbf{3}$.
Correct Answer: 3