Consider the following sequence of numbers: 3, 5, 5, 19, 7, 41, 9, ? Find the number that should replace the question mark so that the pattern of the series is preserved.

Introduction / Context:
This number series alternates between an odd integer and another value related to that odd integer. The task is to recognise how each pair of terms is connected and then extend the same relationship to find the missing final term in the sequence.

Given Data / Assumptions:

• Series: 3, 5, 5, 19, 7, 41, 9, ?
• The terms appear in pairs: (3, 5), (5, 19), (7, 41), (9, ?).
• We must find the last term from the options 71, 61, 79 and 69.

Concept / Approach:
Observe that the first, third, fifth and seventh terms form the simple sequence of odd numbers: 3, 5, 7, 9. Each odd number is followed by a second number that looks close to its square: 3² = 9, 5² = 25 and 7² = 49. However, the actual following terms are 5, 19 and 41, which are each slightly less than these squares. This suggests that the second term in each pair might be given by odd² minus an increasing even number.

Step-by-Step Solution:
Step 1: Pair the sequence as (3, 5), (5, 19), (7, 41), (9, ?). Step 2: For the first pair, 3² = 9 and 9 - 4 = 5. So the second term is 3² - 4. Step 3: For the second pair, 5² = 25 and 25 - 6 = 19. So the second term is 5² - 6. Step 4: For the third pair, 7² = 49 and 49 - 8 = 41. So the second term is 7² - 8. Step 5: The subtracted numbers 4, 6 and 8 form a simple increasing even-number pattern, each increasing by 2. Step 6: Following this logic, for the odd number 9, we should subtract the next even number 10. Step 7: Compute 9² = 81 and then 81 - 10 = 71.

Verification / Alternative check:
The pattern 'odd term, then odd² minus an even number' is consistent across the entire series: 3 → 5, 5 → 19, 7 → 41, all following odd² - 4, odd² - 6, odd² - 8. Extending this to odd² - 10 for 9 gives 71, confirming that 71 is the natural continuation.

Why Other Options Are Wrong:
The values 61, 79 and 69 do not fit the rule odd² - an even number increasing by 2. For 9² = 81, subtracting 20 would give 61, subtracting 2 would give 79 and subtracting 12 would give 69, none of which align with the progression 4, 6, 8, 10. Therefore these options are inconsistent with the pattern and must be rejected.

Common Pitfalls:
Many learners initially search for a single rule linking all eight numbers rather than grouping them into pairs. Another common mistake is to miss the simple sequence of even numbers 4, 6, 8 and instead try more complicated differences. Recognising the alternating structure of the sequence helps simplify the reasoning significantly.

Final Answer:
The missing term that preserves the pattern of odd squares with increasing even subtractions is 71, so option (a) is correct.