The average of 44 consecutive odd numbers is 144. What is the largest odd number in this list of 44 consecutive odd integers?

Introduction / Context:
Here we are dealing with consecutive odd numbers, which form an arithmetic progression with a common difference of 2. The question gives us the average of 44 such numbers and asks for the largest number. This tests understanding of averages and the structure of arithmetic sequences, especially when the count of terms is even.

Given Data / Assumptions:

• There are 44 consecutive odd numbers.
• Their average is 144.
• We need to find the largest number in this sequence.
• The numbers are in increasing order and differ by 2 between consecutive terms.

Concept / Approach:
For any arithmetic sequence, the average is equal to the value of the middle term (if the number of terms is odd) or the average of the two middle terms (if the number of terms is even). Here, 44 is an even number, so the mean is the average of the 22nd and 23rd terms. Let the smallest odd number be a. Then the nth term is a + (n - 1) * 2. We use the mean relation to link the first and last terms.

Step-by-Step Solution:
Let the first term (smallest odd number) be a. Number of terms n = 44. Common difference d = 2. Last term L = a + (44 - 1) * 2 = a + 86. Average of an arithmetic sequence = (first term + last term) / 2. Given average = 144, so: (a + (a + 86)) / 2 = 144. (2a + 86) / 2 = 144. 2a + 86 = 288. 2a = 202. a = 101. Therefore, largest term L = a + 86 = 101 + 86 = 187.
Verification / Alternative check:
First term = 101, last term = 187. Check average: (101 + 187) / 2 = 288 / 2 = 144, which matches the given value. Thus our computed largest number is consistent.
Why Other Options Are Wrong:
Option A (189): Would give average (a + 189) / 2 greater than 144, contradicting the information. Option B (191): Even further from the required mean. Option C (187): Correct largest term obtained from the arithmetic sequence formula. Option D (193): Too large and incompatible with average 144.
Common Pitfalls:
Some students mistakenly assume that the given average is itself one of the sequence terms, which is not always true when there is an even number of terms. Others may forget that the common difference is 2 for odd numbers and use incorrect expressions for the last term. Errors also occur if the formula for the mean of an arithmetic sequence is not applied correctly.
Final Answer:
The largest odd number in the list is 187.