Complete the missing number in the series 5, 20, 76, ?, 448.

Introduction / Context:
This number series question involves a pattern where each term is generated from the previous one using both multiplication and subtraction. The consistent use of these two operations with the same number is the key to solving the problem.

Given Data / Assumptions:

• Series: 5, 20, 76, ?, 448.
• Exactly one middle term is missing.
• The rule should be uniform throughout the series.

Concept / Approach:
We suspect that each term is obtained by multiplying the previous term by some integer and then subtracting that same integer. This is a common pattern in number series questions. We can test this hypothesis using the known terms.

Step-by-Step Solution:
Step 1: From 5 to 20, check if we can express 20 as 5 * k − k for some k. Step 2: Try k = 5: 5 * 5 − 5 = 25 − 5 = 20, which matches the series. So the rule here is multiply by 5 then subtract 5. Step 3: Assume this rule continues: term_{n+1} = term_n * k − k, where k decreases by 1 each time: 5, 4, 3, 2, and so on. Step 4: From 20 to 76, check k = 4: 20 * 4 − 4 = 80 − 4 = 76, which fits. Step 5: For the missing term X, use k = 3: X = 76 * 3 − 3 = 228 − 3 = 225. Step 6: From X to 448, use k = 2: 225 * 2 − 2 = 450 − 2 = 448, which matches the given last term.

Verification / Alternative check:
Reconstructing the entire series with the rule term_{n+1} = term_n * k − k and k values 5, 4, 3, 2 gives: 5, 20, 76, 225, 448. Each step is consistent with the pattern: 5 * 5 − 5 = 20, 20 * 4 − 4 = 76, 76 * 3 − 3 = 225, 225 * 2 − 2 = 448. This confirms that 225 is the correct missing number.

Why Other Options Are Wrong:
Choosing 185, 415, or 232 would break the pattern. For example, if the missing term were 232, then the relation 76 * 3 − 3 = 232 would not hold, and 232 * 2 − 2 would not equal 448. Therefore none of these alternatives maintain a consistent descending k pattern.

Common Pitfalls:
One pitfall is to look for a single fixed multiplier and ignore the possibility that both multiplication and subtraction might be in play. Another is not noticing that the value subtracted is the same as the multiplier, which simplifies the rule significantly. Always test simple combined operations like k times previous minus k or plus k.

Final Answer:
The missing number that completes the series correctly is 225.