Question

Find the arithmetic mean of the first 10 natural numbers.

Answer 1

Hint: Here, to find the arithmetic mean we have the formula:
 $ Mean=\dfrac{Sum\text{ }of\text{ }terms}{Number\text{ }of\text{ }terms} $
We also use that the sum of ‘n’ natural numbers is $ \dfrac{n(n+1)}{2} $ . Here, $ n=10 $ , which is the number of terms.

Complete step-by-step answer:
Here, we have to find the arithmetic of the first 10 natural numbers.
We know that the arithmetic mean to us also the average value.
Hence, the arithmetic mean is defined as:
 $ Mean=\dfrac{Sum\text{ }of\text{ }the\text{ }terms}{Number\text{ }of\text{ }terms} $

Here, the sum of terms is the sum of the first 10 natural numbers.
Therefore the number of terms, n = 10.
 $ mean=\dfrac{1+2+3+4+5+6+7+8+9+10}{10}\text{ }.....\text{ (1)} $
Hence, we can write

First we have to find the sum of the first 10 natural numbers.
We know that the sum of first n natural numbers = $ 1+2+3+...+n $
Since, they are in AP, because the common difference is 1. We can use the formula for sum:
 $ {{S}_{n}}=\dfrac{n}{2}\left( {{a}_{1}}+{{a}_{n}} \right) $
Here, $ {{S}_{n}} $ is the sum of n terms, $ n $ is the number of terms, $ {{a}_{1}} $ is the first term and $ {{a}_{n}} $ is the last term.
Here for the first n terms, $ {{a}_{1}}=1 $ and $ {{a}_{n}}=n $ .
Therefore, we can write:
 $ {{S}_{n}}=\dfrac{n}{2}\left( 1+n \right) $ .

Hence, the sum of first n numbers is $ \dfrac{n(n+1)}{2} $ .
So, the sum of first 10 terms is given by:
 $ \begin{align}
  & {{S}_{10}}=\dfrac{10(10+1)}{2} \\
 & {{S}_{10}}=\dfrac{10\times 11}{2} \\
 & {{S}_{10}}=\dfrac{110}{2} \\
\end{align} $
 By cancellation we get:
 $ {{S}_{10}}=55 $
 Hence the sum of the first 10 terms, $ 1+2+3+...+10=55 $ .

By substituting the value in equation (1) we obtain:
 $ \begin{align}
  & mean=\dfrac{55}{10}\text{ } \\
 & mean=5.5 \\
\end{align} $

Hence, the arithmetic mean of first 10 natural numbers is 5.5

Note: Here, if you know the formula for the sum of first n natural numbers i.e. $ \dfrac{n(n+1)}{2} $ , you can directly substitute and find the answer. Here the number of terms is given, which is 10. Therefore, you can directly substitute in the formula and then find the mean.