The arithmetic mean of 1, 2, 3, ..., n, is

\frac{n - 1}{2}

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\frac{n + 1}{2}

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\frac{n}{2}

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\frac{n}{2} + 1

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Solution

The correct option is A $\frac{n + 1}{2}$
$\Rightarrow$ We have the sequence, $1, 2, 3....... n$
$\Rightarrow$ This is an AP, with the initial term $a = 1$ and the common difference $d = 1$ .
$∴$ The sum of $n$ terms of an AP is given by,
$\Rightarrow$ $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$

$\Rightarrow$ $S_{n} = \frac{n}{2} [2 \times 1 + (n - 1) \times 1]$

$\Rightarrow$ $S_{n} = \frac{n}{2} [2 + (n - 1)]$

$\Rightarrow$ $S_{n} = \frac{n}{2} [n + 1]$
$\to$ Arithmetic mean of $n$ numbers $a_{1}, a_{2}, a_{3}, a_{4}, . . . a_{n}$ is given by the formula
$\Rightarrow$ $A r i t h m e t i c m e a n = \frac{a_{1} + a_{2} + a_{3} + a_{4} + . . . + a_{n}}{n}$

$\Rightarrow$ $A r i t h m e t i c m e a n = \frac{S_{n}}{n}$

$\Rightarrow$ $A r i t h m e t i c m e a n = \frac{\frac{n}{2} [n + 1]}{n}$

$∴$ $A r i t h m e t i c m e a n = \frac{n + 1}{2}$

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