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Let x be the mean of squares of first n natural numbers and y be the square of mean of first n natural numbers. If x/y = 55/42, then what is the value of n ?
- a)24
- b)25
- c)27
- d)30
Correct answer is option 'C'. Can you explain this answer?
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Let x be the mean of squares of first n natural numbers and y be the s...
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
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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Community Answer
Let x be the mean of squares of first n natural numbers and y be the s...
Concept:
- Mean of x1, x2, ....,xn is
- 1 + 2 + 3 + ....+ n =
- 12 + 22 + 32 + ....+ n2 =
Calculation:
Given: x is the mean of squares of first n natural numbers and y is the square of mean of first n natural numbers
⇒ 2 × 42(2n + 1) = 3 × 55(n + 1)
⇒ 168n + 84 = 165n + 165
⇒ 3n = 81
⇒ n = 27
∴ The correct option is (3).
Given: x is the mean of squares of first n natural numbers and y is the square of mean of first n natural numbers
⇒ 2 × 42(2n + 1) = 3 × 55(n + 1)
⇒ 168n + 84 = 165n + 165
⇒ 3n = 81
⇒ n = 27
∴ The correct option is (3).
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