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Difficulty: MediumConsecutive Integers and Number Sets

Set SS consists of all consecutive integers from −m-m to nn, inclusive, where mm and nn are positive integers with n>mn > m. If set SS contains exactly 2525 integers and the sum of all integers in set SS is 7575, what is the value of mm?

Answer: 9

Answer

The value of mm is 99.
To find mm, use the two fundamental properties of consecutive integer sets: term count and set sum. The number of terms from −m-m to nn inclusive is n−(−m)+1=n+m+1=25n - (-m) + 1 = n + m + 1 = 25, which yields n+m=24n + m = 24. The sum of an evenly spaced set is the product of the number of terms and the arithmetic mean of the smallest and largest terms: 25×−m+n2=7525 \times \frac{-m + n}{2} = 75, which simplifies to n−m=6n - m = 6. Subtracting n−m=6n - m = 6 from n+m=24n + m = 24 gives 2m=182m = 18, so m=9m = 9.

Step-by-Step Solution

1
Set up the equation for the number of terms in the set.
n+m=24n + m = 24
The number of integers from −m-m to nn inclusive is n−(−m)+1=n+m+1=25n - (-m) + 1 = n + m + 1 = 25.
2
Set up the equation for the sum of the integers in the set.
n−m=6n - m = 6
The sum of an arithmetic progression is given by number of terms×mean=25×−m+n2=75\text{number of terms} \times \text{mean} = 25 \times \frac{-m + n}{2} = 75, leading to n−m2=3\frac{n - m}{2} = 3.
3
Solve for mm using the two linear equations.
m=9m = 9
Subtracting n−m=6n - m = 6 from n+m=24n + m = 24 yields 2m=182m = 18, giving m=9m = 9.

Key Concept

Properties of consecutive integer sets: inclusive term counting (n−start+1n - \text{start} + 1) and set sum calculation (count×mean\text{count} \times \text{mean}).
Estimated Time:1m 30s
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Topics

Consecutive Integers and Number Sets

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