In the series 625, 625, 600, ?, 475, 875, find the missing number that makes the sequence follow a consistent rule.

Introduction / Context:
This problem presents a relatively short but non obvious number series. The pattern is not a simple arithmetic progression. Instead, it emerges when the numbers are grouped into pairs and the relationship between the pair sums is studied. Our goal is to use this structure to find the missing term.

Given Data / Assumptions:

• Series: 625, 625, 600, ?, 475, 875.
• There are six numbers, which can naturally be thought of as three pairs.
• We assume a regular relationship among these pairs rather than individual successive differences.

Concept / Approach:
When a series has an even number of terms, especially six, it can be useful to group them into pairs: (first, second), (third, fourth) and (fifth, sixth). Then we examine how the sums or differences of these pairs behave. If the sums follow a simple pattern, we can deduce the missing value that restores this pattern.

Step-by-Step Solution:
Step 1: Form pairs from the series: (625, 625), (600, ?), (475, 875).Step 2: Compute the sums of the first and third pairs: 625 + 625 = 1250 and 475 + 875 = 1350.Step 3: Observe that these sums differ by 100: 1350 - 1250 = 100.Step 4: A natural progression for three pair sums is an arithmetic sequence: 1250, 1300, 1350, with a common difference of 50.Step 5: Therefore the middle pair should sum to 1300.Step 6: Since the third term in the original series is 600, we require 600 + ? = 1300. Thus, ? = 1300 - 600 = 700.

Verification / Alternative check:
With the missing term set to 700, the complete pairs become (625, 625), (600, 700) and (475, 875). Their sums are 1250, 1300 and 1350, which form a clean arithmetic progression with common difference 50. No other option yields such a simple and elegant pattern for the pair sums.

Why Other Options Are Wrong:
Values like 545, 675, 725 or 650 would give middle pair sums of 1145, 1275, 1325 or 1250 respectively, none of which produce an evenly spaced sequence between 1250 and 1350. Therefore these alternatives break the emerging structure and cannot be correct.

Common Pitfalls:
A common mistake is to search only for patterns in the single step differences between adjacent terms, which seem irregular here. Recognising that the sequence might be organised into meaningful pairs and looking at pair sums reveals the hidden regularity in this series.

Final Answer:
The missing number that makes the series follow a consistent pair sum pattern is 700.