Find the sum of all even numbers up to and including 1672.

Introduction / Context:Even numbers form the AP: 2, 4, 6, …, 1672. Summing an AP is routine using count and average (or direct sum formula).

Given Data / Assumptions:

• First term 2, last term 1672, d = 2.
• Include 1672 because the prompt says “including it”.

Concept / Approach:Compute n = last/2, then S_n = n/2 * (first + last).

Step-by-Step Solution:n = 1672 / 2 = 836.S = 836/2 * (2 + 1672) = 418 * 1674 = 699,732.

Verification / Alternative check:Equivalently, S = n * average = 836 * (2 + 1672)/2 = 836 * 837 = 699,732.

Why Other Options Are Wrong:699,731 and 699,831/699,832 miss by ±1 or ±100 due to arithmetic slips.

Common Pitfalls:Excluding the endpoint; miscomputing n or (first + last).

Final Answer:699732