Question

Find the mean of the first ten odd natural numbers.
(a) 8
(b) 9
(c) 10
(d) 11

Answer 1

Hint: Apply the formula for the sum of ‘n’ odd natural numbers given by, \[{{S}_{n}}={{n}^{2}}\]. Here, \[{{S}_{n}}\] is the sum of ‘n’ odd natural numbers, ‘n’ is the number of terms. Now, apply the formula for mean given by: - \[\overline{x}=\dfrac{{{S}_{n}}}{n}\], where ‘\[\overline{x}\]’ is the notation of mean. Substitute the value of n = 10 and calculate the answer.

Complete step-by-step solution:
Here, we have to find the mean of the first ten odd natural numbers.
We know that natural numbers are the counting numbers like - 1, 2, 3, ….. up to infinite. Since odd numbers are the numbers which are not divisible by 2. For example, 1, 3, 5, 7, 9, ….. up to infinite. So, combining these two definitions we can conclude that odd natural numbers will be: - 1, 3, 5, 7, …. up to infinite.
Here, we have to consider only 10 odd natural numbers. They will be 1, 3, 5, 7, …. up to 10 terms. Now, to calculate the mean we need to find the sum of these 10 terms.
We know that the sum of ‘n’ odd natural numbers is \[{{n}^{2}}\]. Let us see how. Consider ‘n’ odd natural numbers: -
1, 3, 5, 7, 9, ……, (2n – 1)
Clearly, we can see that the above terms are in A.P. with first term (a) = 1, common difference = 2 and last term (\[{{T}_{n}}\]) = (2n – 1). So, applying the formula for sum of ‘n’ terms of an A.P., we get,
\[\begin{align}
  & \Rightarrow {{S}_{n}}=\dfrac{n}{2}\left[ a+{{T}_{n}} \right] \\
 & \Rightarrow {{S}_{n}}=\dfrac{n}{2}\left[ 1+2n-1 \right] \\
 & \Rightarrow {{S}_{n}}=\dfrac{n}{2}\times 2n \\
 & \Rightarrow {{S}_{n}}={{n}^{2}} \\
\end{align}\]
So, it is proved that the sum of ‘n’ odd natural numbers is \[{{n}^{2}}\]. We have been given only 10 terms, so for n = 10, we have,
\[\Rightarrow {{S}_{10}}={{10}^{2}}=100\] - (i)
Now, applying the formula for the mean, we get,
\[\overline{x}=\dfrac{{{S}_{n}}}{n}\], ‘\[\overline{x}\]’ is notation of mean.
Substituting the value of n = 10 and \[{{S}_{10}}=100\], we get,
\[\begin{align}
  & \Rightarrow \overline{x}=\dfrac{{{S}_{10}}}{10} \\
 & \Rightarrow \overline{x}=\dfrac{100}{10} \\
 & \Rightarrow \overline{x}=10 \\
\end{align}\]
Hence, option (c) is the correct answer.

Note: One may note that one can clear his/her doubt about the formula of the sum of ‘n’ odd natural numbers equal to \[{{n}^{2}}\] by adding the first 10 such numbers. You have to add 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19. You will get the same sum that is 100. Now, remember that here only 10 terms were given therefore we can easily add them without using formula but if a large number of terms are given then we must remember and apply the formula.