Question

Find the mean of the first twelve odd natural numbers.
(a) 10
(b) 11
(c) 12
(d) None

Answer 1

Hint: For solving this problem we use general formula of mean given as
\[\text{mean = }\dfrac{\text{sum of numbers}}{\text{number of numbers}}\]
We need to find the sum of first \['n'\] odd natural numbers by taking the general representation of the add number as \[\left( 2k-1 \right)\]. Then by substituting \[n=12\] we get a sum of numbers so that we can easily calculate the mean.

Complete step-by-step solution:
First let us find the sum of first \['n'\] odd natural numbers.
We know that the general representation of add number is \[\left( 2k-1 \right)\] where \[k=1,2,3.....n\]
So, we can write the sum of these \['n'\] odd natural numbers in summation form as
\[\Rightarrow {{S}_{n}}=\sum\limits_{k=1}^{n}{\left( 2k-1 \right)}\]
By separating the terms in summation we get
\[\Rightarrow {{S}_{n}}=2\sum\limits_{k=1}^{n}{k}-\sum\limits_{k=1}^{n}{1}........equation(i)\]
We know that sum of first \['n'\] natural numbers as \[\sum\limits_{k=1}^{n}{k}=\dfrac{n\left( n+1 \right)}{2}\].
Also,\[\sum\limits_{k=1}^{n}{1}\] means adding ‘1’ by \['n'\] times so, we can write \[\sum\limits_{k=1}^{n}{1}=n\]
By substituting these values in equation (i) we get
\[\begin{align}
  & \Rightarrow {{S}_{n}}=2\left( \dfrac{n\left( n+1 \right)}{2} \right)-n \\
 & \Rightarrow {{S}_{n}}={{n}^{2}}+n-n \\
 & \Rightarrow {{S}_{n}}={{n}^{2}} \\
\end{align}\]
We are asked to find the mean of the first ten odd natural numbers.
By replacing \[n=12\] in above equation we get sum of first 12 odd natural numbers as
\[\Rightarrow {{S}_{12}}={{12}^{2}}=144\]
Let us assume that \[X\] be the mean of first 12 odd natural numbers
We know that the general formula of mean given as
\[\text{mean = }\dfrac{\text{sum of numbers}}{\text{number of numbers}}\]
By substituting the required parameters in mean formulae we get
\[\begin{align}
  & \Rightarrow X=\dfrac{{{S}_{12}}}{n} \\
 & \Rightarrow X=\dfrac{144}{12}=12 \\
\end{align}\]
Therefore, we can say the mean of the first 12 odd natural numbers is 12.
So, option (c) is the correct answer.

Note: This problem can be solved in another method.
We know that the first 12 odd natural numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17, 19,21,23.
Let us assume that \[X\] be the mean of first 12 odd natural numbers
We know that the general formula of mean given as
\[\text{mean = }\dfrac{\text{sum of numbers}}{\text{number of numbers}}\]
By substituting the required parameters in mean formulae we get
\[\begin{align}
  & \Rightarrow X=\dfrac{1+3+5+7+9+11+13+15+17+19+21+23}{12} \\
 & \Rightarrow X=\dfrac{144}{12}=12 \\
\end{align}\]
Therefore, we can say the mean of the first 12 odd natural numbers is 12.
So, option (c) is the correct answer.