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UPSC NDA Math Model Paper 1

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Question : 14 of 120

Marks: +1, -0

What is the mean of first n natural numbers ?

$n$
$\frac{n+1}{2}$
$\frac{n(n+1)}{2}$
None of the above

Solution:

Concept:

Suppose there are 'n' observations

\{x_{1}, x_{2}, x_{3}, \dots, x_{n}\}

Mean

(\overline{x}) = \frac{x_{1}+x_{2}+x_{3}+\dots+x_{n}}{n} = \frac{\sum\limits_{i=1}^{n} x_{i}}{n}

Sum of the natural numbers

\frac{n(n+1)}{2}

Calculation:

To find: Mean of first n natural numbers

As we know, Sum of first n natural numbers

\frac{n(n+1)}{2}

Now, Mean

\frac{\frac{n(n+1)}{2}}{n}

\frac{n+1}{2}

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