Series summation — first 84 even natural numbers Compute the exact sum: 2 + 4 + 6 + … up to 84 terms.

Introduction / Context:
Sums of arithmetic progressions (AP) are common in aptitude exams. The even numbers form an AP with first term 2 and common difference 2. Knowing and applying the AP sum formula makes the calculation quick and reliable.

Given Data / Assumptions:

• Total terms n = 84.
• First term a = 2, common difference d = 2.
• We need S = 2 + 4 + … (84 terms).

Concept / Approach:
The sum of the first n even numbers equals 2*(1 + 2 + … + n). Hence S = 2 * (n*(n+1)/2) = n*(n+1). This compact relation eliminates the need to list terms individually.

Step-by-Step Solution:
Use S = n*(n+1).For n = 84, S = 84*85.Compute 84*85 = (84*80) + (84*5) = 6720 + 420 = 7140.Therefore, the sum is 7140.

Verification / Alternative check:
AP formula: S = n/2 * (2a + (n-1)d) = 84/2 * (4 + 83*2) = 42 * (4 + 166) = 42 * 170 = 7140, confirming the result.

Why Other Options Are Wrong:
7540, 6720, 8832 arise from off-by-one term counts or multiplying 84 by 84 instead of 85.

Common Pitfalls:
Confusing the number of terms with the largest even value; forgetting that the nth even number is 2n; and mishandling 84*85 mental multiplication.

Final Answer:
7140