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Find the arithmetic mean of the first 12 natural numbers.

seo-qna
Last updated date: 13th Jun 2024
Total views: 411.9k
Views today: 7.11k
Answer
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Hint- Arithmetic mean is the ratio of the sum of the observations to the total number of observations. Arithmetic mean gives us an idea about the middle point or center point of the observations through which deviation and error can be determined accordingly.



Complete step by step solution:
The set of numbers lying to the right-hand side of the number line except 0 are known as Natural Numbers. When 0 is included with all the natural numbers then, they will result in the Whole Numbers.

Arithmetic mean is the ratio of the sum of the observations to the total number of observations. Arithmetic mean gives us an idea about the middle point or center point of the observations through which deviation and error can be determined accordingly.

In this question, the arithmetic mean of the first 1 natural numbers needed to be determined for which first we need to determine the sum of the first 12 natural numbers and then divide the sum obtained by the total number of observations ie., 12 to get the desired result.

First 12 natural numbers are: 1,2,3,4,5,6….12.

Let the sum of the first 12 natural numbers be $ S $ then,

$
  S = 1 + 2 + 3 + 4 + 5 + 6 + ...12 \\
   = 78 \\
 $

Now, substitute 78 for the sum of observations and 12 for the total number of observations in the formula $M = \dfrac{S}{n}$ to determine the arithmetic mean.

$
  M = \dfrac{S}{n} \\
   = \dfrac{{78}}{{12}} \\
   = 6.5 \\
 $

Hence, 6.5 is the arithmetic mean of the first 12 natural numbers.

Note: Alternatively, the sum of the first 12 natural numbers can also be found by using the formula of the AP series i.e., $ S = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right] $ where, $n$ is the total number of terms in the series, $a$ is the first term of the series, and $ d $ is the common difference of the series.