Problem 60 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Statement 1 : The variance of first \(n\) odd natural numbers is \(\frac{n^{2}-1}{3}\) Statement 2: The sum of first \(\mathrm{n}\) odd natural number is \(n^{2}\) and the sum of square of first \(n\) odd natural numbers is \(\frac{n\left(4 n^{2}+1\right)}{3}\). Online May 26, 2012] (a) Statement 1 is true, Statement 2 is false. (b) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement \(1 .\) (c) Statement 1 is false, Statement 2 is true. (d) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1 .

Step-by-Step Solution

Answer

(b) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1.

Step 1: Understanding Statement 1

Statement 1 claims that the variance of the first \(n\) odd natural numbers is \(\frac{n^2 - 1}{3}\). We'll first derive the formula for variance.

Step 2: Calculate Mean of Odd Numbers

The first \(n\) odd natural numbers are \(1, 3, 5, \ldots, (2n-1)\). Their sum is \(n^2\) from the formula in Statement 2, so the mean \(\mu\) is \(\frac{n^2}{n} = n\).

Step 3: Calculate Sum of Squares

The sum of the squares of the first \(n\) odd natural numbers is given by Statement 2 as \(\frac{n(4n^2 + 1)}{3}\).

Step 4: Find the Variance

Variance is calculated as \(\text{Variance} = \frac{1}{n} \left( \sum x_i^2 - n\mu^2 \right)\), where \(\mu = n\) from earlier. Substitute the sum of squares to get:\[\text{Variance} = \frac{1}{n} \left( \frac{n(4n^2 + 1)}{3} - n \times n^2 \right) = \frac{n(4n^2 + 1) - 3n^3}{3n} = \frac{n(4n^2+1) - 3n^3}{3n} = \frac{n^4 + n}{3n} = \frac{4n^3 + n - 3n^3}{3n} = \frac{n^2 - 1}{3}.\]

Step 5: Verify Statement 2

Statement 2 states the sum of the first \(n\) odd numbers is \(n^2\) and sum of squares is \(\frac{n(4n^2 + 1)}{3}\). These are both standard formulas and correct, thus Statement 2 is true.

Step 6: Determine the Relation Between Statements

Even though Statement 2 is true and correct, it simply provides the formulas. The relationship that variance calculation requires these formulas does not imply Statement 2 is an explanation for Statement 1. Therefore, if both statements are true, Statement 2 is not a correct explanation for Statement 1.

Key Concepts

Odd Numbers

Odd numbers are fascinating numbers that cannot be evenly divided by 2. When talking about odd numbers, we are referring to the sequence that begins with

• 1
• 3
• 5
• 7
• ... up to \(2n-1\)

Understanding odd numbers is crucial in various mathematical problems, particularly when dealing with sequences of numbers. The first few odd numbers form a pattern that adds successive twos to the previous odd number.
This sequence is foundational in computing sums, variances, and other properties in mathematics. Knowing how to work with odd numbers can significantly simplify mathematical proofs and calculations.

Sum of Squares

The sum of squares of a series of numbers is a critical concept often encountered in mathematics. When calculating the sum of squares for odd numbers, particularly the first n odd numbers, we use the following formula: \[\frac{n(4n^2 + 1)}{3} \]This formula helps us sum the squares of the first few odd numbers like this:

• 1^2
• 3^2
• 5^2
• ... up to (2n-1)^2

Each number is squared, and the results are totaled. This formula simplifies many calculations and allows easy computation of large sets of odd numbers without manual squaring and adding. Having a solid understanding of sum of squares is vital for mathematical proofs and variance calculation.

Mathematical Proof

A mathematical proof is a logical argument that demonstrates the truth of a given statement. Proofs are essential in mathematics because they provide a validated framework for understanding why something is true.
When working on problems involving variance and sequences like odd numbers, a proof helps to break down why solutions are correct. For example, showing that the variance of the first n odd numbers is \[\frac{n^2 - 1}{3}\]relies on using known formulas and arithmetic steps. The process involves:

• Calculating the mean of the sequence
• Finding the sum of squares
• Substituting into the variance formula

An effective proof not only verifies results but also enhances understanding of the underlying principles. This makes mathematical concepts more accessible and actionable.

Mean Calculation

Calculating the mean of a set of numbers determines the average value of that set. For the first n odd numbers, the arithmetic mean is an essential step in calculating variance.
You find the mean (\(\mu\)) by summing all odd numbers in the sequence and dividing by the count of numbers, which is n. For odd numbers like 1, 3, 5, ..., (2n-1), the sum we use is \(n^2\) from statement 2's formula, leading to:

• Mean \(\mu = \frac{n^2}{n} = n\)

This calculation shows how each odd number contributes to the total, and that the average is straightforward for sequences of numbers, easing many subsequent mathematical operations like variance. Understanding mean calculation is crucial for comprehending aggregate properties of numbers effectively.