# What is the mean of first 99 natural numbers ?

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What is the mean of first 99 natural numbers ?
1. 100
2. 50.5
3. 50
4. 99

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Correct Answer - Option 3 : 50

Concept:

Suppose there are ‘n’ observations {${\rm{\;}}{{\rm{x}}_{\rm{1}}},{{\rm{x}}_{{\rm{2\;}}}},{{\rm{x}}_{{\rm{3\;}}}}, \ldots ,{{\rm{x}}_{{\rm{n\;}}}}$}

Mean $\left( {{\rm{\bar x}}} \right) = \frac{{{\rm{\;}}({{\rm{x}}_1} + {{\rm{x}}_{2{\rm{\;}}}} + {{\rm{x}}_{3{\rm{\;}}}} + \ldots + {{\rm{x}}_{{\rm{n\;}}}})}}{{\rm{n}}}$ $= {\rm{\;}}\frac{{\mathop \sum \nolimits_{{\rm{i = 1}}}^{\rm{n}} {{\rm{x}}_{\rm{i}}}}}{{\rm{n}}}$

Sum of the first n natural numbers = $\rm \frac{n(n+1)}{2}$

Calculation:

To find:  Mean of the first 99 natural numbers

As we know, Sum of first n natural numbers = $\rm \frac{n(n+1)}{2}$

Now, Mean = $\rm \dfrac { \frac{99(99+1)}{2}}{99}$

$\rm \frac{(99+1)}{2}=50$