If A = 1 - 10 + 3 - 12 + 5 - 14 + 7 - 16 + ... up to 60 terms, what is the value of A?

Introduction / Context:
This question involves an alternating series of positive and negative terms where the positive and negative terms each follow their own simple pattern. Such problems test your ability to recognize patterns, group terms intelligently, and sum a finite series without writing out every term individually. Handling signs carefully is crucial in series like this.

Given Data / Assumptions:

• The series is A = 1 - 10 + 3 - 12 + 5 - 14 + 7 - 16 + ...
• The pattern continues in the same manner up to 60 terms in total.
• We are asked to find the final value of A after summing all 60 terms.
• The sequence of positive terms is 1, 3, 5, 7, ... (odd numbers).
• The sequence of negative terms is -10, -12, -14, -16, ... (negative even numbers starting from 10).

Concept / Approach:
The key idea is to group the series into pairs, each consisting of one positive odd number and the following negative even number. Each pair has the form (2k - 1) - (8 + 2k) for some integer k. We can simplify each pair algebraically and notice that the result is a constant, independent of k. Once we know how many such pairs appear in 60 terms, we can multiply the value of one pair by the number of pairs to get the total sum.

Step-by-Step Solution:
Write the first few terms with indices to see the pattern clearly. First pair: 1 - 10. Second pair: 3 - 12. Third pair: 5 - 14. Fourth pair: 7 - 16, and so on. Notice that the positive terms are odd numbers: 1, 3, 5, 7, ... which can be written as 2k - 1. The corresponding negative terms are even numbers starting from 10: 10, 12, 14, 16, ... which can be written as 8 + 2k. So a typical pair is (2k - 1) - (8 + 2k) = 2k - 1 - 8 - 2k. This simplifies to -9 for every pair. The series has 60 terms in total, and each pair uses 2 terms, so there are 60 / 2 = 30 pairs. Therefore, A = 30 * (-9) = -270.

Verification / Alternative Check:
We can verify the pattern by explicitly calculating the first few pair sums. For the first pair, 1 - 10 = -9. For the second pair, 3 - 12 = -9. For the third pair, 5 - 14 = -9. For the fourth pair, 7 - 16 = -9. This confirms that every pair contributes -9 to the sum. Since the pattern of odd positives and even negatives continues consistently, and we have a total of 30 such pairs, the total sum must be 30 * (-9) = -270. There is no leftover single term because 60 is an even number.

Why Other Options Are Wrong:

• Option -360: This would correspond to 40 terms of -9 or some miscount of the number of pairs, so it does not match the correct pairing.
• Option -310: This is not a multiple of -9 and therefore cannot result from 30 identical pair sums of -9.
• Option -240: This equals -8 * 30, which would only occur if each pair summed to -8, which is not the case here.
• Option -300: This equals -10 * 30 and could result from a mistaken assumption that each pair sums to -10, but actual computations show that each pair sums to -9.

Common Pitfalls:
Students often try to handle such series term by term instead of recognizing the repeating pattern in pairs. This can lead to arithmetic errors or confusion. Another common issue is misidentifying the general form of the terms and thereby computing the pair sum incorrectly. Some may forget to divide the total number of terms by 2 to get the number of pairs, leading to an incorrect multiplier. Careful identification of the pair structure and verification of the pattern with the first few pairs helps avoid these mistakes.

Final Answer:
The value of A for the given 60 term series is -270.