The average of 44 consecutive odd numbers is 144. Based on this information, what is the largest odd number in this consecutive sequence?

Introduction / Context:
This question uses the properties of arithmetic progressions and odd numbers. You are told that the average of 44 consecutive odd numbers is 144 and you must determine the largest number in the sequence. Because consecutive odd numbers differ by 2, they form a regular arithmetic progression, and the mean is directly related to the first and last terms.

Given Data / Assumptions:
- We have 44 consecutive odd numbers.
- Average (mean) of these 44 numbers = 144.
- The numbers are odd and consecutive, so the common difference is 2.
- We need to find the largest odd number in this sequence.

Concept / Approach:
For any arithmetic progression, the average of all terms equals the average of the first and last terms, that is (first + last) / 2. For an odd count of terms, this is also the middle term, but here the count is even (44 terms). However, the relation between first and last still holds. If we denote the first term as a and the last term as l, then (a + l) / 2 = 144 and l can be expressed in terms of a and the number of terms using the common difference. Solving these equations yields both a and l.

Step-by-Step Solution:
Step 1: Let the first odd number in the sequence be a and the last (largest) number be l.Step 2: For an arithmetic progression, average = (a + l) / 2. Given average is 144, so (a + l) / 2 = 144.Step 3: Thus a + l = 288.Step 4: Since the numbers are consecutive odd numbers, the common difference is 2 and the number of terms n = 44.Step 5: For an arithmetic sequence, l = a + (n - 1) * d, where d is common difference.Step 6: Here l = a + (44 - 1) * 2 = a + 43 * 2 = a + 86.Step 7: Substitute l into a + l = 288: a + (a + 86) = 288.Step 8: This gives 2a + 86 = 288, so 2a = 202 and a = 101.Step 9: Now l = a + 86 = 101 + 86 = 187.Step 10: Therefore, the largest number in the sequence is 187.

Verification / Alternative check:
As a quick check, the first odd number is 101 and the last is 187. The midpoint between these two is (101 + 187) / 2 = 288 / 2 = 144, which matches the given average. The number of terms from 101 to 187 with step 2 is (187 - 101) / 2 + 1 = 86 / 2 + 1 = 43 + 1 = 44, so both the count and the average are consistent.

Why Other Options Are Wrong:
Values like 189, 191 or 193 would result in a different average when paired with the corresponding first term. For example, if the largest number were 189, the first number would adjust to keep the average 144, but the resulting sequence might not have exactly 44 terms. Only 187 produces both the required average and the correct number of consecutive odd terms.

Common Pitfalls:
Students sometimes confuse the formula for the nth term with the formula for average, or they accidentally use a common difference of 1 instead of 2 for odd numbers. Others forget that there are 44 terms, not 43. Keeping track of the common difference and the number of terms is essential for a correct answer.

Final Answer:
The largest odd number in the sequence is 187.