What is the value of the sum 91 + 92 + 93 + ... + 140, that is, the total of all integers from 91 to 140 inclusive?

Introduction / Context:
This question asks you to find the sum of a sequence of consecutive integers from 91 to 140. Such sums form an arithmetic progression, and there is a simple formula to compute them without adding each term individually. Recognizing and using this formula saves time and reduces errors in exams.

Given Data / Assumptions:
First term of the sequence = 91.Last term of the sequence = 140.All integers between these two values, including both endpoints, are included.We must find the total sum of these 50 integers.

Concept / Approach:
The numbers from 91 to 140 form an arithmetic progression with common difference 1. For an arithmetic sequence, the sum S of n terms with first term a and last term l is given by S = n * (a + l) / 2. The key steps are to determine the number of terms n, then substitute into this formula and simplify carefully.

Step-by-Step Solution:
Step 1: Compute the number of terms n from 91 to 140 inclusive.Step 2: Use n = last term − first term + 1 = 140 − 91 + 1.Step 3: Calculate n = 49 + 1 = 50.Step 4: Use the sum formula for an arithmetic sequence: S = n * (first term + last term) / 2.Step 5: Substitute a = 91, l = 140 and n = 50: S = 50 * (91 + 140) / 2.Step 6: Compute the numerator inside the brackets: 91 + 140 = 231.Step 7: Then S = 50 * 231 / 2 = 25 * 231.Step 8: Multiply 231 by 25: 231 * 25 = 231 * (100 / 4) = (23100 / 4) = 5775.

Verification / Alternative check:
Another way is to pair numbers from opposite ends. Pair 91 with 140, 92 with 139, 93 with 138 and so on. Each pair sums to 231. Because there are 50 numbers, there are 25 pairs. So the total sum is 25 * 231, which again equals 5775. This agrees with the formula method and confirms the result.

Why Other Options Are Wrong:
The value 11550 is exactly double the correct sum and might result from forgetting to divide by 2 in the formula. The larger values 17325 and 23100 would correspond to summing a much larger range of numbers or making more severe arithmetic errors. None of them are consistent with correctly applying the arithmetic series formula.

Common Pitfalls:
Some learners miscount the number of terms and use n = 49 instead of 50. Others forget the division by 2 in the sum formula or add the endpoints incorrectly. Carefully computing n and stepping through the formula helps avoid these mistakes.

Final Answer:
The value of the sum 91 + 92 + ... + 140 is 5775.