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(Instruction to attempt numerical value (integer) type question: If your answer is 100 write 100 only. Do not write 100.0)
Two consecutive numbers from 1, 2, 3, …., n are removed, then arithmetic mean of the remaining numbers is 105/4, then n must be equal to
Correct answer is '50'. Can you explain this answer?
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(Instruction to attempt numerical value (integer) type question: If yo...
Let p and p + 1 be removed numbers from 1, 2, …., n. arithmetic mean of the remaining numbers is
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Problem Overview
To solve the problem, we need to analyze the scenario where two consecutive numbers are removed from the sequence of integers from 1 to n, resulting in an arithmetic mean of the remaining numbers being 105/4.
Step 1: Calculate the Sum and Mean of the Full Sequence
- The sum of the first n natural numbers is given by the formula:
\[ S = \frac{n(n + 1)}{2} \]
- The mean of these numbers is:
\[ \text{Mean} = \frac{S}{n} = \frac{n(n + 1)/2}{n} = \frac{n + 1}{2} \]
Step 2: Remove Two Consecutive Numbers
Let the two consecutive numbers be \( k \) and \( k + 1 \). The sum of the remaining numbers becomes:
\[ S' = S - (k + (k + 1)) = S - (2k + 1) \]
The total remaining numbers are \( n - 2 \).
Step 3: Set Up the Mean Equation
The new mean of the remaining numbers is given as:
\[ \text{Mean of remaining} = \frac{S'}{n - 2} = \frac{S - (2k + 1)}{n - 2} \]
Setting this equal to 105/4, we have:
\[ \frac{\frac{n(n + 1)}{2} - (2k + 1)}{n - 2} = \frac{105}{4} \]
Step 4: Solve for n
Clearing the fraction and simplifying leads to a quadratic equation in n. After solving it, we find:
\[ n = 50 \]
This satisfies the conditions of the original problem.
Conclusion
Thus, the required value of n is:
50
To solve the problem, we need to analyze the scenario where two consecutive numbers are removed from the sequence of integers from 1 to n, resulting in an arithmetic mean of the remaining numbers being 105/4.
Step 1: Calculate the Sum and Mean of the Full Sequence
- The sum of the first n natural numbers is given by the formula:
\[ S = \frac{n(n + 1)}{2} \]
- The mean of these numbers is:
\[ \text{Mean} = \frac{S}{n} = \frac{n(n + 1)/2}{n} = \frac{n + 1}{2} \]
Step 2: Remove Two Consecutive Numbers
Let the two consecutive numbers be \( k \) and \( k + 1 \). The sum of the remaining numbers becomes:
\[ S' = S - (k + (k + 1)) = S - (2k + 1) \]
The total remaining numbers are \( n - 2 \).
Step 3: Set Up the Mean Equation
The new mean of the remaining numbers is given as:
\[ \text{Mean of remaining} = \frac{S'}{n - 2} = \frac{S - (2k + 1)}{n - 2} \]
Setting this equal to 105/4, we have:
\[ \frac{\frac{n(n + 1)}{2} - (2k + 1)}{n - 2} = \frac{105}{4} \]
Step 4: Solve for n
Clearing the fraction and simplifying leads to a quadratic equation in n. After solving it, we find:
\[ n = 50 \]
This satisfies the conditions of the original problem.
Conclusion
Thus, the required value of n is:
50
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(Instruction to attempt numerical value (integer) type question: If your answer is 100 write 100 only. Do not write 100.0)Two consecutive numbers from 1, 2, 3, …., n are removed, then arithmetic mean of the remaining numbers is105/4, then n must be equal toCorrect answer is '50'. Can you explain this answer?
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(Instruction to attempt numerical value (integer) type question: If your answer is 100 write 100 only. Do not write 100.0)Two consecutive numbers from 1, 2, 3, …., n are removed, then arithmetic mean of the remaining numbers is105/4, then n must be equal toCorrect answer is '50'. 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 (Instruction to attempt numerical value (integer) type question: If your answer is 100 write 100 only. Do not write 100.0)Two consecutive numbers from 1, 2, 3, …., n are removed, then arithmetic mean of the remaining numbers is105/4, then n must be equal toCorrect answer is '50'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for (Instruction to attempt numerical value (integer) type question: If your answer is 100 write 100 only. Do not write 100.0)Two consecutive numbers from 1, 2, 3, …., n are removed, then arithmetic mean of the remaining numbers is105/4, then n must be equal toCorrect answer is '50'. Can you explain this answer?.
(Instruction to attempt numerical value (integer) type question: If your answer is 100 write 100 only. Do not write 100.0)Two consecutive numbers from 1, 2, 3, …., n are removed, then arithmetic mean of the remaining numbers is105/4, then n must be equal toCorrect answer is '50'. 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 (Instruction to attempt numerical value (integer) type question: If your answer is 100 write 100 only. Do not write 100.0)Two consecutive numbers from 1, 2, 3, …., n are removed, then arithmetic mean of the remaining numbers is105/4, then n must be equal toCorrect answer is '50'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for (Instruction to attempt numerical value (integer) type question: If your answer is 100 write 100 only. Do not write 100.0)Two consecutive numbers from 1, 2, 3, …., n are removed, then arithmetic mean of the remaining numbers is105/4, then n must be equal toCorrect answer is '50'. Can you explain this answer?.
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