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Difficulty: MediumConsecutive Integers and Number Sets

Set SS consists of nn consecutive integers. The arithmetic mean of the 55 smallest integers in Set SS is −12-12, and the arithmetic mean of the 55 largest integers in Set SS is 2424. What is the value of nn?

  1. A
    36
  2. B
    37
  3. C
    40
  4. 41Answer
  5. E
    42

Answer

41
For an evenly spaced set of 5 consecutive integers, the arithmetic mean equals the middle term. Therefore, the 3rd smallest term of Set SS is −12-12, making the 1st term x=−14x = -14. Similarly, the 3rd term from the end of Set SS is 2424. Representing the 3rd term from the end as x+n−3x + n - 3 and substituting x=−14x = -14 yields −14+n−3=24-14 + n - 3 = 24, which simplifies to n=41n = 41.

Step-by-Step Solution

1
Express the 5 smallest integers and find the first term of the set.
Let the set SS be represented as {x,x+1,x+2,…,x+n−1}\{x, x+1, x+2, \dots, x+n-1\}. The 5 smallest integers are x,x+1,x+2,x+3,x+4x, x+1, x+2, x+3, x+4. Their arithmetic mean is the middle term, x+2x+2. Setting x+2=−12x+2 = -12 yields x=−14x = -14.
In any set of consecutive integers with an odd number of elements, the arithmetic mean equals the median (middle term).
2
Express the 5 largest integers and set up an equation for nn.
The 5 largest integers in Set SS are x+n−5,x+n−4,x+n−3,x+n−2,x+n−1x+n-5, x+n-4, x+n-3, x+n-2, x+n-1. Their arithmetic mean is the middle term, x+n−3x+n-3. Setting x+n−3=24x+n-3 = 24 and substituting x=−14x = -14 gives −14+n−3=24-14 + n - 3 = 24.
The 5 largest elements also form an evenly spaced set whose mean is the middle of those 5 terms.
3
Solve for nn.
n−17=24  ⟹  n=41n - 17 = 24 \implies n = 41.
Simplifying the linear equation gives the total number of consecutive integers in Set SS.

Key Concept

Arithmetic Mean and Median Equivalence in Consecutive Integer Subsets
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Topics

Consecutive Integers and Number Sets

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