Let P = 2^2 + 6^2 + 10^2 + 14^2 + ... + 94^2 and Q = 1^2 + 5^2 + 9^2 + ... + 81^2, where each sequence is an arithmetic progression of integers whose squares are summed. What is the value of P - Q?

Introduction / Context:
This question deals with sums of squares of terms in arithmetic progressions. Two sequences are given: one starting at 2 and increasing by 4 up to 94, and the other starting at 1 and increasing by 4 up to 81. Instead of summing all the squares directly, we are asked to find the difference P - Q between the two total sums. Problems of this sort test understanding of patterns, arithmetic progressions and efficient computation strategies, which are important in aptitude and quantitative reasoning exams.

Given Data / Assumptions:

• P = 2^2 + 6^2 + 10^2 + ... + 94^2.
• Q = 1^2 + 5^2 + 9^2 + ... + 81^2.
• Both sequences have common difference 4.
• All terms are integers and the series are finite.
• We are required to compute P - Q exactly.

Concept / Approach:
One approach is to treat each series as a separate sum of squares of an arithmetic progression and use formulae. Another, often simpler, approach for exam style questions is to compute P and Q systematically or pair terms and look for patterns. Because the upper and lower bounds are moderate, we can think of each term as (4k + 2)^2 or (4k + 1)^2 and sum them over appropriate ranges. Then we subtract Q from P to get the final answer. Using symbolic or careful arithmetic avoids mistakes.

Step-by-Step Solution:
Step 1: Observe that P contains terms of the form (4k + 2)^2 from 2 to 94. For k = 0,1,2,...,23, the term is 2,6,10,...,94, so there are 24 terms. Step 2: Similarly, Q contains terms of the form (4k + 1)^2 from 1 to 81. For k = 0,1,2,...,20, the term is 1,5,9,...,81, so there are 21 terms. Step 3: Compute P = Σ (4k + 2)^2 for k = 0 to 23. Each term expands to 16k^2 + 16k + 4. Step 4: Compute Q = Σ (4k + 1)^2 for k = 0 to 20. Each term expands to 16k^2 + 8k + 1. Step 5: Use standard summation formulas for Σk and Σk^2 or compute the sums carefully to obtain P and Q. Step 6: After performing the arithmetic (or using a systematic calculation), we obtain P - Q = 26075.

Verification / Alternative check:
As a verification step, one can compute P and Q using a spreadsheet or a programmable calculator by listing each term, squaring it and summing the results. Doing so yields the same final difference P - Q = 26075. Because both sequences and all operations are straightforward and purely arithmetic, agreement between two independent methods is a strong confirmation that the result is correct.

Why Other Options Are Wrong:
The other options 24645, 29317, 31515 and 30500 are all plausible looking large numbers but do not match the exact arithmetic. Any small error in how many terms are included or in the formulas for the sums of squares would lead to one of these incorrect values. Only 26075 is consistent with an exact computation of both sums and their difference.

Common Pitfalls:
A common error is miscounting the number of terms, for example assuming there are 23 rather than 24 terms in P, or 22 instead of 21 terms in Q. Another common mistake is forgetting that the numbers are squares, not just the original arithmetic progression values. Errors in expanding (4k + c)^2 or in using the sum of first n integers and sum of squares formulas can also lead to incorrect results. Carefully writing out the general term and the index range before summing helps avoid these mistakes.

Final Answer:
Therefore, the value of P - Q is 26075.