Question

The mean of first $n$ odd natural numbers is given as $\dfrac{{{n}^{2}}}{81}$. What is $n$?
(a) 9
(b) 81
(c) 27
(d) None of these

Answer 1

We will first find the formula for sum of the first $n$ odd natural numbers. To find such a formula, we will use the formula for the sum of an arithmetic progression. Using this formula, we will give the formula for the mean of first $n$ odd natural numbers. Then, we will use the given information regarding the mean and the formula we gave for mean to form an equation. Solving this equation will give us the value of $n$.

Complete step by step answer:
The first $n$ odd natural numbers are $1,3,5,\ldots ,\left( 2n-1 \right)$. We can see that this is an arithmetic progression with first term as 1 and the common difference as 2. We know that the sum of $n$ terms of an arithmetic progression is given by the following formula,
$\text{sum of first n terms = }\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)$
where $a$ is the first term and $d$ is the common difference. Using this formula, the sum of the first $n$ odd natural numbers will be calculated as follows,
$\text{sum of first n odd natural numbers = }\dfrac{n}{2}\left( 2\times 1+\left( n-1 \right)\times 2 \right)$
Simplifying the above expression we get
$\begin{aligned} & \text{sum of first n odd natural numbers = }\dfrac{n}{2}\left( 2+2n-2 \right) \\\ & =\dfrac{n}{2}\times 2n \\\ & ={{n}^{2}} \end{aligned}$
So the sum of the first $n$ odd natural numbers is ${{n}^{2}}$. Now, the mean of the first $n$ odd natural numbers will be,
$\text{mean of the first n odd natural numbers = }\dfrac{\text{sum of the first n odd natural numbers}}{n}$
So, we get $\text{mean of the first n odd natural numbers = }\dfrac{{{n}^{2}}}{n}=n$.
We are given that the mean of first $n$ odd natural numbers is $\dfrac{{{n}^{2}}}{81}$. So, from the above expression, we get that
$\dfrac{{{n}^{2}}}{81}=n$
Simplifying this expression, we get $n=81$.
Hence, the correct option is (b).
Note:
It is essential that we realize the first $n$ odd natural numbers as an arithmetic progression. Since we know the formula for the sum of any general arithmetic progression, it becomes easier to find the sum of the first $n$ odd natural numbers. It is important that we write the expressions and values explicitly. This will be useful in avoiding any minor mistakes in the calculations.