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The average of n consecutive numbers is ' a '.If any one number is deleted from these ' n ' numbers then the average of the remaining ( n -1 ) numbers cannot be less than?
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The average of n consecutive numbers is ' a '.If any one number is del...
Understanding the Average of Consecutive Numbers
When dealing with n consecutive numbers, the average is defined as 'a'. This average represents the central tendency of these numbers, which can be expressed mathematically.
Calculating the Total Sum
- The total sum of these n consecutive numbers can be calculated as:
Total Sum = n * a
Effect of Deleting One Number
When one number is deleted from the set of n numbers, the average of the remaining (n - 1) numbers is affected.
New Average Calculation
- Let the deleted number be 'x'.
- The sum of the remaining (n - 1) numbers will be:
Remaining Sum = Total Sum - x = n * a - x
- The new average of the remaining numbers is:
New Average = (Remaining Sum) / (n - 1) = (n * a - x) / (n - 1)
Minimum Average Condition
To find the minimum possible average of the remaining numbers after deleting one number, consider:
- If 'x' is the maximum number in the original set, the average will be minimized.
- The maximum number in n consecutive numbers can be expressed as:
Max Number = a + (n - 1) * d (where d = 1 for consecutive integers)
Thus, the average of the remaining (n - 1) numbers cannot be less than:
- New Average ≥ (n * a - (a + (n - 1))) / (n - 1)
This leads to:
Conclusion
The average of the remaining (n - 1) numbers cannot be less than 'a' - 0.5, ensuring that the average remains relatively close to the original average 'a'.
When dealing with n consecutive numbers, the average is defined as 'a'. This average represents the central tendency of these numbers, which can be expressed mathematically.
Calculating the Total Sum
- The total sum of these n consecutive numbers can be calculated as:
Total Sum = n * a
Effect of Deleting One Number
When one number is deleted from the set of n numbers, the average of the remaining (n - 1) numbers is affected.
New Average Calculation
- Let the deleted number be 'x'.
- The sum of the remaining (n - 1) numbers will be:
Remaining Sum = Total Sum - x = n * a - x
- The new average of the remaining numbers is:
New Average = (Remaining Sum) / (n - 1) = (n * a - x) / (n - 1)
Minimum Average Condition
To find the minimum possible average of the remaining numbers after deleting one number, consider:
- If 'x' is the maximum number in the original set, the average will be minimized.
- The maximum number in n consecutive numbers can be expressed as:
Max Number = a + (n - 1) * d (where d = 1 for consecutive integers)
Thus, the average of the remaining (n - 1) numbers cannot be less than:
- New Average ≥ (n * a - (a + (n - 1))) / (n - 1)
This leads to:
Conclusion
The average of the remaining (n - 1) numbers cannot be less than 'a' - 0.5, ensuring that the average remains relatively close to the original average 'a'.
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The average of n consecutive numbers is ' a '.If any one number is deleted from these ' n ' numbers then the average of the remaining ( n -1 ) numbers cannot be less than?
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The average of n consecutive numbers is ' a '.If any one number is deleted from these ' n ' numbers then the average of the remaining ( n -1 ) numbers cannot be less than? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The average of n consecutive numbers is ' a '.If any one number is deleted from these ' n ' numbers then the average of the remaining ( n -1 ) numbers cannot be less than? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The average of n consecutive numbers is ' a '.If any one number is deleted from these ' n ' numbers then the average of the remaining ( n -1 ) numbers cannot be less than?.
The average of n consecutive numbers is ' a '.If any one number is deleted from these ' n ' numbers then the average of the remaining ( n -1 ) numbers cannot be less than? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The average of n consecutive numbers is ' a '.If any one number is deleted from these ' n ' numbers then the average of the remaining ( n -1 ) numbers cannot be less than? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The average of n consecutive numbers is ' a '.If any one number is deleted from these ' n ' numbers then the average of the remaining ( n -1 ) numbers cannot be less than?.
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