Which number replaces the question mark in the series: 108, 127, 2413, 2430, 41310, ?, 619875?

Introduction / Context:
This is a more challenging mixed operation series where each step alternates between addition and multiplication, and the numbers used for these operations themselves follow a descending odd number pattern. It tests both calculation skills and the ability to spot structured alternation in a complex sequence.

Given Data / Assumptions:

• Series: 108, 127, 2413, 2430, 41310, ?, 619875.
• Exactly one middle term is missing.
• The operations between terms are likely to alternate between addition and multiplication using related numbers.

Concept / Approach:
We examine how to obtain each successive term from the one before it, testing a pattern of adding an odd number and then multiplying by another related odd number. With some experimentation, we find that the odd numbers used reduce in a regular fashion, creating a descending sequence like 19, 17, 15 in the operations.

Step-by-Step Solution:
Step 1: From 108 to 127, the increase is 127 - 108 = 19. Step 2: From 127 to 2413, compute 127 × 19 = 2413. Thus the pair of operations between these two terms are +19, then ×19. Step 3: From 2413 to 2430, the increase is 2430 - 2413 = 17. Step 4: From 2430 to 41310, compute 2430 × 17 = 41310. So we have +17, then ×17. Step 5: We see a clear pattern: starting from 108, add 19 then multiply by 19; then add 17 and multiply by 17. The odd numbers used in this pair of operations are decreasing by 2 each cycle. Step 6: The next pair of operations should use the odd number 15. So from 41310 we add 15, then multiply by 15. Step 7: First compute the missing term as 41310 + 15 = 41325. Step 8: Then check the last term: 41325 × 15 = 619875, which matches the final value in the given series.

Verification / Alternative check:
Summarise the pattern: 108 (+19) 127 (×19) 2413 (+17) 2430 (×17) 41310 (+15) 41325 (×15) 619875. The odd numbers used for both addition and multiplication step down by 2 each time: 19, 17, 15. This provides a very neat and self confirming structure, so there is no ambiguity about the missing term being 41325.

Why Other Options Are Wrong:
If we choose 41975, 84375, or 41365 and then attempt to apply the multiply by 15 rule, the resulting value will not be 619875, so the given final term would no longer fit the pattern. Moreover, these alternatives also break the clean “add 15, then multiply by 15” step that continues the descending odd number scheme initiated earlier in the sequence.

Common Pitfalls:
At first glance, many students try simple difference or ratio methods and quickly get lost in the large numbers. The key is to notice the use of the same odd number for both the addition and the subsequent multiplication at each stage. Another pitfall is to assume a constant operation across the entire series instead of an alternating one, which does not work here.

Final Answer:
The number that correctly replaces the question mark in the series is 41325.