Examkin

Question

Difficulty: HardConsecutive Integers and Number Sets

A set SS consists of consecutive integers. The arithmetic mean of all the positive integers in set SS is 18.518.5, and the arithmetic mean of all the negative integers in set SS is −12-12. How many integers are in set SS?

Answer: 60

Answer

The total number of integers in set SS is 6060.
The positive integers in set SS are 1,2,…,361, 2, \dots, 36, which have an arithmetic mean of 1+362=18.5\frac{1 + 36}{2} = 18.5. The negative integers in set SS are −23,−22,…,−1-23, -22, \dots, -1, which have an arithmetic mean of −23+(−1)2=−12\frac{-23 + (-1)}{2} = -12. Since set SS consists of consecutive integers spanning from −23-23 to 3636, it includes 2323 negative integers, 3636 positive integers, and the integer 00. The total number of elements is 23+1+36=6023 + 1 + 36 = 60.

Step-by-Step Solution

1
Find the largest positive integer in set SS.
The largest positive integer is 3636, meaning there are 3636 positive integers in SS.
The positive integers in SS must form a consecutive sequence starting at 11 up to some maximum integer mm. The average of consecutive integers from 11 to mm is 1+m2\frac{1 + m}{2}. Setting 1+m2=18.5\frac{1 + m}{2} = 18.5 yields 1+m=371 + m = 37, so m=36m = 36.
2
Find the smallest negative integer in set SS.
The smallest negative integer is −23-23, meaning there are 2323 negative integers in SS.
The negative integers in SS must form a consecutive sequence ending at −1-1 down to some minimum integer −k-k. The average of consecutive integers from −k-k to −1-1 is −k+(−1)2\frac{-k + (-1)}{2}. Setting −k−12=−12\frac{-k - 1}{2} = -12 yields −k−1=−24-k - 1 = -24, so k=23k = 23.
3
Calculate the total number of elements in set SS.
Set SS contains 6060 integers.
Because set SS contains consecutive integers ranging from negative to positive values, it must also contain 00. The total count is 23 (negative integers)+1 (the integer zero)+36 (positive integers)=6023\text{ (negative integers)} + 1\text{ (the integer zero)} + 36\text{ (positive integers)} = 60.

Key Concept

Evenly spaced set averages and classification of zero in consecutive integer sets.
Estimated Time:2m 0s
Rate this question

Topics

Consecutive Integers and Number Sets

More questions from this topic