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If the mean of the squares of first n natural numbers be 11, then n is equal to
- a)13
- b)5
- c)- 13/2
- d)11
Correct answer is option 'B'. Can you explain this answer?
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If the mean of the squares of first n natural numbers be 11, then n is...
**Given information:**
The mean of the squares of the first n natural numbers is 11.
**To find:**
The value of n.
**Solution:**
Let's first calculate the sum of the squares of the first n natural numbers.
The sum of the squares of the first n natural numbers can be calculated using the formula:
Sum = n(n+1)(2n+1)/6
Now, we are given that the mean of these squares is 11.
Mean = Sum/n
Substituting the value of Sum, we get:
11 = n(n+1)(2n+1)/6n
Simplifying further, we get:
11 = (n+1)(2n+1)/6
Multiplying both sides by 6, we get:
66 = (n+1)(2n+1)
Expanding the equation, we get:
66 = 2n^2 + 3n + n + 1
Rearranging the equation, we get:
2n^2 + 4n - 65 = 0
Now, we can solve this quadratic equation for n using factorization or the quadratic formula.
By factoring, we can write:
2n^2 + 4n - 65 = 0
(n + 13)(2n - 5) = 0
So, n can be either -13/2 or 5.
Since n represents the number of natural numbers, it cannot be a negative or fractional value. Therefore, the only valid solution is n = 5.
Hence, the correct answer is option B.
The mean of the squares of the first n natural numbers is 11.
**To find:**
The value of n.
**Solution:**
Let's first calculate the sum of the squares of the first n natural numbers.
The sum of the squares of the first n natural numbers can be calculated using the formula:
Sum = n(n+1)(2n+1)/6
Now, we are given that the mean of these squares is 11.
Mean = Sum/n
Substituting the value of Sum, we get:
11 = n(n+1)(2n+1)/6n
Simplifying further, we get:
11 = (n+1)(2n+1)/6
Multiplying both sides by 6, we get:
66 = (n+1)(2n+1)
Expanding the equation, we get:
66 = 2n^2 + 3n + n + 1
Rearranging the equation, we get:
2n^2 + 4n - 65 = 0
Now, we can solve this quadratic equation for n using factorization or the quadratic formula.
By factoring, we can write:
2n^2 + 4n - 65 = 0
(n + 13)(2n - 5) = 0
So, n can be either -13/2 or 5.
Since n represents the number of natural numbers, it cannot be a negative or fractional value. Therefore, the only valid solution is n = 5.
Hence, the correct answer is option B.
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If the mean of the squares of first n natural numbers be 11, then n is equal toa)13b)5c)- 13/2d)11Correct answer is option 'B'. Can you explain this answer?
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If the mean of the squares of first n natural numbers be 11, then n is equal toa)13b)5c)- 13/2d)11Correct answer is option 'B'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If the mean of the squares of first n natural numbers be 11, then n is equal toa)13b)5c)- 13/2d)11Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the mean of the squares of first n natural numbers be 11, then n is equal toa)13b)5c)- 13/2d)11Correct answer is option 'B'. Can you explain this answer?.
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