Problem 55 - A Complete Resource Book in Mathematics for JEE Main

Question

The variance of the first 50 even natural numbers is (A) \(\frac{833}{4}\) (B) 833 (C) 437 (D) \(\frac{437}{4}\)

Step-by-Step Solution

Answer

The variance is 833 (Option B).

Step 1: Identify the Sequence

Recognize that the first 50 even natural numbers are an arithmetic sequence starting from 2 and ending at 100. The sequence is: 2, 4, 6, ..., 100.

Step 2: Calculate the Mean

The arithmetic mean of an arithmetic sequence can be calculated by the formula \( \text{Mean} = \frac{a + l}{2} \), where \(a\) is the first term and \(l\) is the last term. Here, \(a = 2\) and \(l = 100\). Therefore, \( \text{Mean} = \frac{2 + 100}{2} = 51 \).

Step 3: Calculate the Variance

The variance of an arithmetic sequence with common difference \(d\) is given by \( \text{Variance} = \frac{(n^2 - 1)d^2}{12} \), where \(n\) is the number of terms. Substituting \(n = 50\) and \(d = 2\), we get \( \text{Variance} = \frac{(50^2 - 1) \times 2^2}{12} = \frac{(2500 - 1) \times 4}{12} = \frac{2499 \times 4}{12} = \frac{9996}{12} = 833. \)

Key Concepts

First 50 Even Natural Numbers

To understand the sequence of the first 50 even natural numbers, we begin with the smallest even number which is 2. Even numbers are those numbers that are divisible by 2 without leaving a remainder. The next number after 2 following this rule is 4. Thus, the pattern for even numbers becomes clear: 2, 4, 6, 8, and so on. This is an arithmetic sequence where each term is simply 2 more than the previous one. To find the 50th even natural number, recall that the nth even natural number is given by the formula: \(a_n = 2n\). Therefore, if \(n=50\), the 50th even number is \(2 \times 50 = 100\). Accordingly, the sequence of the first 50 even natural numbers is \(2, 4, 6, \ldots, 100\). This sequence features a very crucial aspect, which is that it is evenly incremented by the common difference of 2, which is vital when calculating variance and other statistical measures.

Arithmetic Mean Calculation

Finding the arithmetic mean of a sequence is a fundamental concept in statistics. To find the mean of an arithmetic sequence, such as the sequence of the first 50 even natural numbers, we can use the formula: \[ \text{Mean} = \frac{a + l}{2} \]Here, \(a\) is the first term and \(l\) is the last term. So for our sequence, \(a = 2\) and \(l = 100\). Plugging in these values, we find: \[ \text{Mean} = \frac{2 + 100}{2} = \frac{102}{2} = 51 \]This result shows that the average of all 50 even natural numbers from 2 to 100 is 51. The arithmetic mean represents the central value of the sequence, essentially acting as a balance point where all the numbers could pivot equally.

Variance Formula for Sequences

Variance is a measure of dispersion that tells us how far the values in a dataset are spread out from the arithmetic mean. In the case of the first 50 even natural numbers, we use a special variance formula intended for arithmetic sequences:\[ \text{Variance} = \frac{(n^2 - 1)d^2}{12} \]In this formula:

• \(n\) is the number of terms in the sequence, which is 50.
• \(d\) is the common difference between each term, which in our case is 2.

Substituting the respective values we have:\[ \text{Variance} = \frac{(50^2 - 1) \times 2^2}{12} = \frac{2499 \times 4}{12} = \frac{9996}{12} = 833 \]Therefore, the variance of the sequence is 833. This tells us that there is a considerable spread in the values of the first 50 even numbers, indicating that the values diverge moderately from the mean of 51. Understanding the variance helps us comprehensively assess the nature and behavior of numbers within the sequence.