Sequences & Series
Non-terminating mean deviation of consecutive integers
MJAT_TS8_P1
Grade 12
Question:
Let $n$ be a natural number with $10\leq n\leq 1000$. The mean deviation of $n$ consecutive integers about the median is $\mu$. If $\mu$ is a non-terminating decimal and also rational, let $\lambda$ be the count of such $n$. The value of $\dfrac{\lambda}{82}$ is:
Step-by-Step Solution
Key Concept: For odd $n$: $\mu=\frac{n^2-1}{4n}=\frac{n}{4}-\frac{1}{4n}$. This is non-terminating iff $4n$ has a prime factor other than 2 or 5 (i.e., $n$ has such a factor). For even $n=2k$: $\mu=n/4$ is always terminating. Count odd $n\in[10,1000]$ with $\mu$ non-terminating rational.
$\lambda/82=\mathbf{6}$.
Correct Answer: 6