Evaluate the sum of the consecutive integers: Compute the value of the series 51 + 52 + 53 + ... + 100.

Introduction:
This question involves finding the sum of a sequence of consecutive integers. Recognizing this as an arithmetic series allows us to use the standard formula for the sum of the first n terms of an arithmetic progression (A.P.), making the calculation quick and reliable.

Given Data / Assumptions:

• We must find S = 51 + 52 + 53 + ... + 100.
• The terms increase by 1 each time, so this is an arithmetic progression.
• We assume all operations are performed on exact integers.

Concept / Approach:
For an arithmetic progression with first term a, last term l, and number of terms n, the sum is given by: S = n * (a + l) / 2. Our task is to determine a, l, and n for this series. The numbers from 51 to 100 are consecutive, so finding n is only a matter of counting how many integers lie in that closed interval.

Step-by-Step Solution:
First term a = 51. Last term l = 100. Number of terms n = (last − first) + 1 = (100 − 51) + 1. Compute n: 100 − 51 = 49, and 49 + 1 = 50, so n = 50. Use the arithmetic series sum formula: S = n * (a + l) / 2 = 50 * (51 + 100) / 2. Compute a + l = 51 + 100 = 151. So S = 50 * 151 / 2. Simplify: 50 / 2 = 25, so S = 25 * 151. Now compute 25 * 151: 25 * 150 = 3750 and 25 * 1 = 25, total 3750 + 25 = 3775. Therefore, the sum S is 3775.

Verification / Alternative check:
Another way is pairing terms: (51 + 100), (52 + 99), (53 + 98), and so on. Each pair sums to 151. Since there are 50 terms total, there are 25 such pairs. Sum = 25 * 151 = 3775, confirming our previous result.

Why Other Options Are Wrong:
The options 2525, 2975, and 3225 are typical incorrect results that might come from miscounting the number of terms or miscomputing the average. None of these match the exact calculation using the correct arithmetic series formula.

Common Pitfalls:
Students often forget to add 1 when computing the number of terms using (last − first) + 1. Another frequent error is to mistakenly use the average of the first few terms instead of the first and last term to form the mean. Carefully applying the standard formula avoids these mistakes.

Final Answer:
The value of the sum 51 + 52 + 53 + ... + 100 is 3775.