The variance of first n even natural numbers is $\frac{n^2 - 1}{4}$.
The sum of first n natural numbers is $\frac{n(n + 1)}{2}$ and the sum of squares of first n natural numbers is $\frac{n(n + 1)(2n +1)}{6}$.
- Statement-1 is true, Statement-2 is true Statement-2 is not a correct explanation for Statement-1
- Statement-1 is true, Statement-2 is false
- Statement-1 is false, Statement-2 is true
- Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for statement -1
The Correct Option is C
Solution and Explanation
Given that;
Sum of first ‘n’ even natural numbers
\(=2+4+6+.....+2n\)
\(=2(1+2+.....n)\)
\(\frac{2(n+1)n}{2}=n(n+1)\)
For the numbers 2, 4, 6, 8, ......., 2n
\(\bar{x} = \frac{2\left[n\left(n+1\right)\right]}{2n} =\left(n+1\right)\)
And \(Var = \frac{\sum\left(x -\bar{x}\right)^{2}}{2n} = \frac{\sum x^{2}}{n} - \left(\bar{x}\right)^{2}\)
\(= \frac{4\sum n^{2}}{n} - \left(n+1\right)^{2}= \frac{4n\left(n+1\right)\left(2n+1\right)}{6n} - \left(n+1\right)^{2}\)
\(= \frac{2\left(2n+1\right)\left(n+1\right)}{3} - \left(n+1\right)^{2}= \left(n+1\right) \left[\frac{4n+2-3n-3}{3} \right]\)
\(= \frac{\left(n+1\right)\left(n-1\right)}{3} = \frac{n^{2}-1}{3}\)
\(\therefore\) Statement-1 is false. Clearly, statement - 2 is true .
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".