Difficulty: Easy
Correct Answer: 23
Explanation:
Introduction / Context:
This problem deals with consecutive integers and their average. For an odd number of consecutive terms, the average is exactly equal to the middle term. Once we identify the middle term, it is very easy to reconstruct the full sequence and find the largest value.
Given Data / Assumptions:
Concept / Approach:
For any set of consecutive integers with an odd count, the average is the middle element. If the 7 numbers are:
n1, n2, n3, n4, n5, n6, n7
then n4 is the middle number because there are 3 numbers before it and 3 numbers after it. Hence the average equals n4. We then step up and down by 1 to get the remaining numbers, and identify the largest.
Step-by-Step Solution:
Step 1: Since there are 7 consecutive integers, the middle term is the 4th term.
Step 2: Average = 20, so the middle term n4 = 20.
Step 3: The sequence is:
n1 = 20 - 3 = 17,
n2 = 20 - 2 = 18,
n3 = 20 - 1 = 19,
n4 = 20,
n5 = 20 + 1 = 21,
n6 = 20 + 2 = 22,
n7 = 20 + 3 = 23.
Step 4: Largest integer in the sequence = n7 = 23.
Verification / Alternative check:
Check average: sum = 17 + 18 + 19 + 20 + 21 + 22 + 23.
17 + 18 = 35, +19 = 54, +20 = 74, +21 = 95, +22 = 117, +23 = 140.
Average = 140 / 7 = 20, which matches the given condition.
Why Other Options Are Wrong:
Options A (21) and B (22) are values in the sequence but are not the largest term.
Option D (24) and E (25) are larger but would force the average to be greater than 20 if they were part of a 7-term consecutive sequence centered at 20.
Common Pitfalls:
Confusing the average with one of the end terms rather than the middle term can lead to an incorrect sequence.
Some students forget that there are 3 steps of +1 to go from the middle to the largest term when there are 7 consecutive numbers.
Final Answer:
The largest of the seven consecutive integers is 23.
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