Find the sum of all even natural numbers between 1 and 31.

Given data

• Even numbers from 2 to 30 inclusive form an arithmetic progression (AP).

Concept / Approach

• AP sum: S = n/2 (first + last); here, difference d = 2.

Step-by-step calculation

First term a = 2; last term l = 30; d = 2Number of terms n = ((l − a)/d) + 1 = ((30 − 2)/2) + 1 = 14 + 1 = 15Sum S = n/2 × (a + l) = 15/2 × 32 = 15 × 16 = 240

Verification

Pairwise: (2+30), (4+28), … each = 32; 15/2 pairs ⇒ 15 × 16 = 240.

Common pitfalls

• Accidentally including 32 or excluding 2.

Final Answer

240