Question

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

Answer 1

$\boxed{\boxed{\mathsf{\frac{n + 1}{2}}}}$
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Let's take $\mathsf{1,2,3........n}$ as an $\textbf{A.P}$

Here,
$\mathsf{a = 1}$
$\mathsf{d = 1}$
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$\underline{\textsf{Using the formula,}}$

$\mathbf{S_n = \frac{n}{2}(2a + (n-1)d)}$
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$\underline{\textsf{Putting the values of a; d in the Formula,}}$

=> $\mathsf{\frac{n}{2}(2(1) + (n-1)1)}$
=> $\mathsf{\frac{n}{2}(2 + n - 1)}$
=>$\mathsf{\frac{n}{2}(n + 1)}$
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Now,

$\mathbf{Mean = \frac{Sum \: of \: Terms}{Total \: Number \: of \: Terms}}$
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=> $\mathsf{\frac{1 + 2 + 3.....a_n}{n}}$

=> $\mathsf{\frac{S_n}{n}}$

Putting the Value of $\mathsf{S_n}$ , we get

=> $\mathsf{\frac{\frac{n}{2}(n + 1)}{n}}$

=> $\boxed{\mathsf{\frac{n + 1}{2}}}$

Hence, the arithmetic mean of 1,2,3.....n is $\mathbf{\frac{n + 1}{2}}$
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Answer 2

Answer:

The arithmetic mean of 1,2,3.......n is equal to n(n+1)/2.

Step-by-step explanation:

Arithmetic mean is equal to sum of the given numbers divided by the total count of numbers in the data.

           Arithmetic mean  =$=\frac{x_{1} +x_{2} +x_{3} .......+x_{n} }{n}$

Now  for the collection of numbers 1,2,3.......n:

             Arithmetic mean   $=\frac{n(n+1)}{2}$