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Difficulty: MediumConsecutive Integers and Number Sets

Set SS consists of a sequence of consecutive integers, ordered from least to greatest. If Set SS contains exactly 4545 integers and the sum of all the integers in Set SS is 405405, what is the value of the smallest integer in Set SS?

Answer: -13

Answer

The smallest integer in Set SS is −13-13.
The average (arithmetic mean) of the set is calculated by dividing the total sum (405405) by the number of terms (4545), which equals 99. In an evenly spaced set with an odd number of elements, the mean is equal to the median (middle term). Because there are 4545 terms, exactly 2222 terms lie below the median. Subtracting 2222 from 99 yields −13-13 as the smallest integer in the set.

Step-by-Step Solution

1
Calculate the arithmetic mean and median of the set
Arithmetic mean = Median = 9
For an evenly spaced set, the arithmetic mean is equal to the median. Dividing the sum (405405) by the number of terms (4545) gives 99.
2
Determine the number of terms preceding the median
22 terms precede the median
With 4545 terms in total, the median is the 23rd23\text{rd} term, which leaves 45−12=22\frac{45 - 1}{2} = 22 terms smaller than the median.
3
Calculate the smallest integer
Smallest integer = −13-13
Subtracting 2222 from the median gives 9−22=−139 - 22 = -13.

Key Concept

In any set of consecutive integers with nn terms, the arithmetic mean equals the median. If nn is odd, the smallest integer is given by Median−n−12\text{Median} - \frac{n-1}{2}.
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Topics

Consecutive Integers and Number Sets

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