Series summation — first 37 odd natural numbers Find the exact sum: 1 + 3 + 5 + … (37 terms).

Introduction / Context: A well-known identity states that the sum of the first n odd numbers equals n^2. This elegant result bypasses lengthy addition and is widely used in number theory and aptitude problems.

Given Data / Assumptions:

• We seek S = 1 + 3 + 5 + … (37 terms).
• Each term increases by 2 (an AP of odds).
• Use the identity for the sum of the first n odd numbers.

Concept / Approach: Identity: sum of first n odd numbers = n^2. For n = 37, compute 37^2 directly. This avoids the AP sum formula steps and leads to an exact integer quickly.

Step-by-Step Solution:Let n = 37.Compute n^2: 37^2 = (40 - 3)^2 = 1600 - 240 + 9 = 1369.So S = 1369.

Verification / Alternative check: Using AP sum: S = n/2 * (first + last). The 37th odd is 2*37 - 1 = 73. Then S = 37/2 * (1 + 73) = 37/2 * 74 = 37 * 37 = 1369, same result.

Why Other Options Are Wrong:1295, 1388, 1875 do not equal 37^2 and come from miscounting terms or miscomputing 37^2.

Common Pitfalls: Using the largest term (73) as n; mixing odds with evens; arithmetic slips in the square computation. Remember the identity: n odds sum to n^2.

Final Answer: 1369