All questions of Measures of Central Tendency for Commerce Exam

Given:
Let x be the mean of squares of the first n natural numbers.
Let y be the square of the mean of the first n natural numbers.
It is given that x/y = 55/42.

To Find:
The value of n.

Explanation:
Let's start by finding the values of x and y.

Finding the value of x:
The squares of the first n natural numbers are 1^2, 2^2, 3^2, ..., n^2.
The sum of these squares can be expressed as:
x = 1^2 + 2^2 + 3^2 + ... + n^2

Using the formula for the sum of squares, we can rewrite this as:
x = n(n + 1)(2n + 1)/6

Finding the value of y:
The mean of the first n natural numbers is the sum of the numbers divided by n.
The sum of the first n natural numbers can be expressed as:
sum = 1 + 2 + 3 + ... + n

Using the formula for the sum of an arithmetic series, we can rewrite this as:
sum = n(n + 1)/2

The mean is sum/n, so we can write the mean as:
mean = n(n + 1)/2n = (n + 1)/2

The square of the mean is:
y = (mean)^2 = [(n + 1)/2]^2 = (n + 1)^2/4

Calculating x/y:
Now, we can calculate x/y using the values we found for x and y:
x/y = (n(n + 1)(2n + 1)/6) / ((n + 1)^2/4)
Simplifying this expression:
x/y = (4n(n + 1)(2n + 1)) / (6(n + 1)^2)
x/y = (4n(2n + 1)) / (6(n + 1))
x/y = (2n(2n + 1)) / (3(n + 1))

Given that x/y = 55/42, we can set up the equation:
(2n(2n + 1)) / (3(n + 1)) = 55/42

Solving the equation:
Cross multiplying:
42 * 2n(2n + 1) = 55 * 3(n + 1)
84n(2n + 1) = 165(n + 1)
168n^2 + 84n = 165n + 165
168n^2 - 81n - 165 = 0

Factoring the quadratic equation:
(8n - 11)(21n + 15) = 0

Setting each factor to zero and solving for n:
8n - 11 = 0 or 21n + 15 = 0
8n = 11 or 21n = -15
n = 11/8 or n = -15/21

Since n represents the number of natural numbers, it cannot be negative. Therefore, n = 11/8 is not a valid solution.

Conclusion:
The valid solution for n

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Question Analysis:
The question asks for the median of the series of all the even terms from 4 to 296. The series includes all the even numbers between 4 and 296. We need to find the middle value of this series, which is the median.

Step-by-Step Solution:
To find the median of the series, we need to follow these steps:

Step 1: List all the even terms from 4 to 296:
The even terms between 4 and 296 are: 4, 6, 8, 10, 12, ..., 296.

Step 2: Calculate the total number of terms in the series:
To find the median, we need to know the total number of terms in the series. In this case, we have a sequence of even numbers, so the total number of terms can be calculated using the formula:
Number of terms = (Last Term - First Term) / Common Difference + 1.

In this case, the first term is 4, the last term is 296, and the common difference is 2 (since we are dealing with even numbers). So, the number of terms = (296 - 4) / 2 + 1 = 146.

Step 3: Find the middle term:
Since we have an odd number of terms (146), the median will be the middle term. To find the middle term, we divide the total number of terms by 2 and round up to the nearest whole number. In this case, the middle term is the 73rd term.

Step 4: Find the value of the middle term:
To find the value of the middle term, we can use the formula for the nth term of an arithmetic sequence:
nth term = first term + (n - 1) * common difference.

In this case, the first term is 4, the common difference is 2, and the value of n is 73. So, the value of the middle term is 4 + (73 - 1) * 2 = 4 + 144 = 148.

Step 5: Determine the median:
The median is the middle value of the series, which is 148.

Conclusion:
The median of the series of all the even terms from 4 to 296 is 148. Therefore, the correct answer is option C) 150.

The Correct Option is (C)
decile divide any series into 10 parts.

For calculating median it is required to set the data either in ascending order or descending order.