The average of 25 consecutive odd integers is 55. What is the largest (highest) integer among these 25 odd numbers?

Introduction / Context:
This question examines your understanding of averages and arithmetic progressions, especially sequences of consecutive odd integers. When a set of numbers is evenly spaced, such as consecutive odd numbers, the average has a direct relationship to the middle term. By using this property, you can find the highest term quickly without listing all numbers.

Given Data / Assumptions:

• There are 25 consecutive odd integers.
• The average of these 25 numbers is 55.
• We must find the largest number in this sequence.
• The numbers form an arithmetic progression with common difference 2.

Concept / Approach:
For an arithmetic progression, the average of the numbers is equal to the value of the middle term when the number of terms is odd. Since we have 25 terms, which is odd, the average 55 is the 13th term in the ordered list. Once we know the middle term, we can move outward to find the largest term by adding the common difference repeatedly. The common difference between consecutive odd integers is 2, so the highest term will be a certain number of steps above the middle term.

Step-by-Step Solution:
Step 1: Recognize that 25 consecutive odd numbers form an arithmetic progression with 25 terms and common difference 2. Step 2: Since there are 25 terms, the middle term is the 13th term. Step 3: In any arithmetic progression with an odd number of terms, the average equals the middle term. Step 4: The given average is 55, so the 13th term in the list is 55. Step 5: There are 12 terms after the 13th term and 12 terms before it. Step 6: Each successive term increases by 2 because the numbers are consecutive odd integers. Step 7: Therefore, the highest term is the 13th term plus 12 steps of size 2. Step 8: Compute the increase: 12 * 2 = 24. Step 9: Add this to the middle term: 55 + 24 = 79. Step 10: Hence, the largest integer among the 25 consecutive odd integers is 79.

Verification / Alternative check:
An alternative method is to consider the first term. Since the 13th term is 55, and each term differs by 2, the first term is 12 steps below 55, with each step subtracting 2. So the first term is 55 − 24 = 31. This gives the sequence 31, 33, 35, up to 79. You can quickly confirm that this sequence has 25 terms and that the average of the first and last term is (31 + 79) / 2 = 55, which matches the given average, verifying the solution.

Why Other Options Are Wrong:
Values like 105, 155, or 109 would correspond to sequences with much higher averages than 55, since the middle term would also increase. The option 75 is close but would correspond to a different average. For example, if 75 were the highest term in a similar sequence, the middle term would be lower, and the resulting average would not be 55. Only 79 fits all the conditions of being the highest term in a sequence of 25 consecutive odd integers with average 55.

Common Pitfalls:
A common mistake is to misinterpret the average as the average of the first and last terms rather than the middle term in an odd length sequence. While the average of all terms is equal to the average of the first and last term, ignoring the connection to the middle term can make the reasoning less direct. Another error is to miscount the number of steps from the middle to the last term, leading to an off by one error. Always count carefully and remember that with 25 terms, there are 12 terms on each side of the middle term.

Final Answer:
The highest among the 25 consecutive odd integers is 79.