Evaluate the sum 14^3 + 16^3 + 18^3 + ... + 30^3.

Introduction / Context:
This problem involves summing cubes of even integers in an arithmetic progression. It tests knowledge of series, in particular the formula for the sum of cubes, and the ability to transform a sum over even numbers into a more manageable form.

Given Data / Assumptions:

• We need to compute S = 14^3 + 16^3 + 18^3 + ... + 30^3.
• The numbers 14, 16, 18, ..., 30 form an arithmetic progression with common difference 2.
• All calculations are exact integer computations.

Concept / Approach:
One useful observation is that each term is the cube of an even number, so we can factor out powers of 2. Specifically, (2k)^3 = 8k^3. This allows us to rewrite the sum over even numbers as a constant multiple of a sum over consecutive integers. We then use the standard formula for the sum of cubes: 1^3 + 2^3 + ... + n^3 = [n(n + 1)/2]^2.

Step-by-Step Solution:
Write each term as the cube of an even integer: 14 = 2*7, 16 = 2*8, ..., 30 = 2*15. Therefore, 14^3 = (2*7)^3 = 8*7^3, 16^3 = 8*8^3, ..., 30^3 = 8*15^3. So S = 8(7^3 + 8^3 + ... + 15^3). Now consider T = 1^3 + 2^3 + ... + 15^3 and U = 1^3 + 2^3 + ... + 6^3. Then 7^3 + 8^3 + ... + 15^3 = T - U. Using the sum of cubes formula, T = [15*16 / 2]^2 = (120)^2 = 14400. Similarly, U = [6*7 / 2]^2 = (21)^2 = 441. Thus 7^3 + 8^3 + ... + 15^3 = 14400 - 441 = 13959. Now S = 8 * 13959 = 111672.

Verification / Alternative check:
We can directly compute each cube and add: 14^3 = 2744, 16^3 = 4096, 18^3 = 5832, 20^3 = 8000, 22^3 = 10648, 24^3 = 13824, 26^3 = 17576, 28^3 = 21952, 30^3 = 27000. Adding these step by step yields 111672, confirming the formula based result.

Why Other Options Are Wrong:
134576, 120212, 115624 and 125000 arise from partial sums, misapplied formulas or arithmetic errors when adding cubes. Because each cube is quite large, even a small mistake in one term can push the total toward one of these incorrect options.

Common Pitfalls:
A common mistake is to confuse the sum of cubes formula with the sum of squares formula, or to forget to subtract the first few cubes when using T - U. Another error is incorrectly computing cubes of two digit numbers. Using the factorisation (2k)^3 = 8k^3 and the standard sum of cubes formula keeps the work organised and reliable.

Final Answer:
The value of 14^3 + 16^3 + 18^3 + ... + 30^3 is 111672.