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Zorluk: OrtaConsecutive Integers and Number Sets

Set SS consists of consecutive integers. The sum of all positive integers in set SS is 300300, and the median of set SS is −4.5-4.5. How many negative integers are in set SS?

Cevap: 33

Cevap

The set S contains 33 negative integers.
To find the number of negative integers in set S, first determine the maximum positive integer N in the set using the sum formula for consecutive positive integers: \frac{N(N+1)}{2} = 300, which yields N = 24. Next, use the property that for an evenly spaced set, the median is the average of the smallest and largest numbers: -4.5 = \frac{\text{Smallest} + 24}{2}. Solving for the smallest integer gives -33. Finally, count the negative integers in set S, which run from -33 to -1 inclusive: -1 - (-33) + 1 = 33.

Adım Adım Çözüm

1
Find the largest integer in Set S
The largest integer is 24.
The positive integers in set S must form a sequence from 1 to N. The sum of positive integers is given by \frac{N(N + 1)}{2} = 300, which simplifies to N(N + 1) = 600. Since 24 \times 25 = 600, N = 24.
2
Find the smallest integer in Set S using the median formula
The smallest integer is -33.
In any set of consecutive integers, the median is equal to the arithmetic mean of the smallest and largest terms: \text{Median} = \frac{\text{Smallest} + \text{Largest}}{2}. Substituting -4.5 for the median and 24 for the largest term gives -4.5 = \frac{\text{Smallest} + 24}{2} \implies \text{Smallest} + 24 = -9 \implies \text{Smallest} = -33.
3
Count the number of negative integers in Set S
There are 33 negative integers.
The negative integers in set S range from -33 to -1, inclusive. The count is calculated as -1 - (-33) + 1 = 33.

Anahtar Kavram

Properties of consecutive integer sets, median-mean equivalence, and inclusive range counting
Tahmini Süre:1m 30s
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Konular

Consecutive Integers and Number Sets

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