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
- $\{99\}$
- $\phi$
- $N$
- $\{9\}$
The Correct Option is C
Solution and Explanation
let be any natural number
Mean deviation about mean
So, it is true for every natural no. ' '
Top Questions on Series
- Let \[ S=\frac{1}{2!5!}+\frac{1}{3!2!3!}+\frac{1}{5!2!1!}+\cdots \text{ up to 13 terms}. \] If $13S=\dfrac{2^k}{n!}$, $k\in\mathbb{N}$, then $n+k$ is equal to
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$\frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \cdots = \frac{\pi^4}{90},$
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$\frac{1}{2^4} + \frac{1}{4^4} + \frac{1}{6^4} + \cdots = \beta.$
Then find $\frac{\alpha}{\beta}$
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Concepts Used:
Arithmetic Progression
Arithmetic Progression (AP) is a mathematical series in which the difference between any two subsequent numbers is a fixed value.
For example, the natural number sequence 1, 2, 3, 4, 5, 6,... is an AP because the difference between two consecutive terms (say 1 and 2) is equal to one (2 -1). Even when dealing with odd and even numbers, the common difference between two consecutive words will be equal to 2.
In simpler words, an arithmetic progression is a collection of integers where each term is resulted by adding a fixed number to the preceding term apart from the first term.
For eg:- 4,6,8,10,12,14,16
We can notice Arithmetic Progression in our day-to-day lives too, for eg:- the number of days in a week, stacking chairs, etc.
Read More: Sum of First N Terms of an AP