Which number should complete the series 0, 1, 4, 15, ?, 325, 1956 so that the same pattern continues throughout?

Introduction / Context:
This question involves a recursive number series where each term is generated from the previous term using multiplication and addition. Such patterns are common in higher level reasoning sections and require you to observe not only the numbers themselves but also how each new term is produced.

Given Data / Assumptions:
- The given series is: 0, 1, 4, 15, ?, 325, 1956.- The pattern is likely based on operations applied to the immediately preceding term.- We assume a consistent rule is applied at every step of the series.

Concept / Approach:
When the series grows quickly but not by fixed multiples, a good strategy is to check if each term equals the previous term multiplied by some integer and then adjusted by the same integer. That is, we test patterns like a(n+1) = a(n) * k + k where k changes in a simple way, such as 1, 2, 3, and so on. Once this structure is identified, we can extend it to find the missing value.

Step-by-Step Solution:
- From 0 to 1: 0 * 1 + 1 = 1.- From 1 to 4: 1 * 2 + 2 = 4.- From 4 to 15: 4 * 3 + 3 = 15.- So we see the pattern a(n+1) = a(n) * (n+1) + (n+1), starting with n = 0.- For the next term after 15, use multiplier 4: 15 * 4 + 4 = 60 + 4 = 64.- Then check the next steps: 64 * 5 + 5 = 320 + 5 = 325.- Next: 325 * 6 + 6 = 1950 + 6 = 1956, which matches the series.

Verification / Alternative check:
- Verify each applied step to ensure no arithmetic mistake: 0, 1, 4, 15, 64, 325, 1956.- The rule a(n+1) = a(n) * (n+1) + (n+1) holds perfectly for every transition.- This confirms that 64 is the only consistent value for the missing term.

Why Other Options Are Wrong:
- 60: This is 15 * 4 but without the +4 part, so it breaks the observed pattern.- 75 and 80: Neither of these can be written as 15 * 4 + 4, and they do not allow the next given terms 325 and 1956 to be generated consistently.- Only 64 maintains the series rule and matches all subsequent terms.

Common Pitfalls:
Students often try to use simple differences or ratios between non-consecutive terms and miss the recursive relationship. Another mistake is to notice only the multiplication and forget the addition part, leading to answers like 60 instead of 64. Always confirm your spotted rule against multiple consecutive steps, not just one or two transitions.

Final Answer:
The number that correctly completes the series is 64, so the correct option is 64.