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Olympiad workbook

Exam: Olympiad workbook
Subject: IOQM
Chapter: Functions

Olympiad workbook · IOQM · Functions

Let n be a positive integer such that 1 ≤ n ≤ 1000 . Let M_n be the...

Let

n

be a positive integer such that

1 \leq n \leq 1000

. Let

M_n

be the number of integers in the set

X_n=\{\sqrt{4 n+1}, \sqrt{4 n+2}, \ldots ., \sqrt{4 n+1000}\}

. Let

a=\max \left\{M_n: 1 \leq n \leq 1000\right\}

and

b=\min \left\{M_n: 1 \leq n \leq 1000\right\}

. Find

a-b

.

Correct answer

22

Step-by-step solution

The spacing of square is less when

n

is greatest and spacing of square is more when

n

is less then

\begin{aligned} & a=M_1 \text { and } b=M_{1000} \\ & \therefore a=29 \text { and } b=7 \\ & \therefore a-b=29-7=22 \end{aligned}

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