Sequences & Series
Integer Answer
MMTS_Full_Test_22
Grade 12
Question:
Let $a_n=\left(1-\dfrac{1}{\sqrt{2}}\right)\cdots\left(1-\dfrac{1}{\sqrt{n+1}}\right)$ for $n\ge 1$. If $k$ is the smallest natural number such that $a_k<\dfrac{1}{10}$, then $k+1$ equals
Step-by-Step Solution
Key Concept: $a_n=\prod_{j=2}^{n+1}(1-1/\sqrt{j})$; find when product $<1/10$
After computation: $k=5$ approximately, $k+1=6$.
Correct Answer: 6