The arithmetic mean of 1 2 3 n is


Asked by admin @ in Math viewed by 287 People


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

Answered by admin @



\boxed{\boxed{\mathsf{\frac{n + 1}{2}}}}
___________________________________

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

Here,
\mathsf{a = 1}
\mathsf{d = 1}
___________________________________

\underline{\textsf{Using the formula,}}

\mathbf{S_n = \frac{n}{2}(2a + (n-1)d)}
___________________________________

\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)}
___________________________________

Now,

\mathbf{Mean = \frac{Sum \: of \: Terms}{Total \: Number \: of \: Terms}}
___________________________________

=> \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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