Question

Let $S$ be the set of all values of $a_1$ for which the mean deviation about the mean of $100$ consecutive positive integers $a_1, a_2, a_3, \ldots , a_{100}$ is $25$ Then $S$ is

• A. $\\{99\\}$
• B. $\phi$
• C. $N$
• D. $\\{9\\}$

Answer 1

Answer: (C)

$N$

Answer 2

The correct answer is (C) : $N$
let a1​ be any natural number
a1​,a1​+1,a1​+2,….,a1​+99 are values of ai​′S
xˉ=100a1​+(a1​+1)+(a1​+2)+…..+a1​+99​
=100100a1​+(1+2+…..+99)​=a1​+2×10099×100​
=a1​+299​
Mean deviation about mean =100i=1∑100​∣xi​−x∣​
=1002(299​+297​+295​+….+21​)​
=1001+3+….+99​
=100250​[1+99]​
=25
So, it is true for every natural no. ' a1​ '