The sum of four consecutive odd integers is 2160. The greatest of them is

Concept

For any set of consecutive terms in arithmetic progression, the terms are evenly spaced about their mean. Four consecutive odd integers have a common difference of 2, so if their sum is S, their mean is S ÷ 4, and the four terms sit symmetrically around that mean as (mean − 3), (mean − 1), (mean + 1), (mean + 3). The greatest term is therefore mean + 3.

Application

1. Let the four consecutive odd integers be x, x + 2, x + 4, x + 6, where x is the smallest.

2. Write the sum condition: x + (x + 2) + (x + 4) + (x + 6) = 2160.

3. Combine like terms: 4x + 12 = 2160.

4. Isolate the variable: 4x = 2148, so x = 537 (the smallest term).

5. The question asks for the GREATEST term, which is x + 6 = 537 + 6 = 543.

Cross-check

Using the mean shortcut: mean = 2160 ÷ 4 = 540, so the four terms are 537, 539, 541, 543 and the greatest is 540 + 3 = 543. Adding back: 537 + 539 + 541 + 543 = 2160 confirms the set. The greatest of the four integers is 543.