In the number sequence 7, 16, 36, 78, 144, ?, each term follows a special pattern based on products of consecutive prime numbers. Which number should replace the question mark?

Introduction / Context:
This problem involves recognizing a pattern in a series of numbers that does not follow a simple arithmetic or geometric rule. Instead, the terms can be expressed using products of prime numbers with a constant adjustment. Such questions are designed to test numerical pattern recognition, understanding of prime numbers and algebraic manipulation in simple expressions like a*b + 1.

Given Data / Assumptions:

- The given sequence is 7, 16, 36, 78, 144, ? - We must identify the rule that generates each term. - The options are 168, 196, 222 and 256. - We assume there is a single consistent pattern that fits all given terms.

Concept / Approach:
A good strategy is to express each term in different forms: differences, ratios or in terms of products of small integers. When we look at the numbers carefully, they can be written as products of prime pairs plus one. For example, 7 = 2*3 + 1 and 16 = 3*5 + 1. The set of multipliers turns out to be consecutive primes. Once we see this, we can extend the pattern by using the next pair of consecutive primes and adding 1 again.

Step-by-Step Solution:
Step 1: Express each term minus one and factor it. Step 2: For 7, compute 7 − 1 = 6 = 2*3, a product of consecutive primes 2 and 3. Step 3: For 16, compute 16 − 1 = 15 = 3*5, again a product of consecutive primes 3 and 5. Step 4: For 36, compute 36 − 1 = 35 = 5*7, consecutive primes 5 and 7. Step 5: For 78, compute 78 − 1 = 77 = 7*11, consecutive primes 7 and 11. Step 6: For 144, compute 144 − 1 = 143 = 11*13, consecutive primes 11 and 13. Step 7: The pattern is clear: each term is p*q + 1 where p and q are consecutive primes: (2,3), (3,5), (5,7), (7,11), (11,13). Step 8: The next consecutive prime pair is 13 and 17. Step 9: Compute 13*17 = 221, then add 1 to get 221 + 1 = 222.

Verification / Alternative check:
Check that using 222 as the next term keeps the pattern intact. Subtract 1: 222 − 1 = 221, which factors as 13*17. This continues the exact rule of using two consecutive primes multiplied together and then adding 1. None of the other options produce such a clean representation with consecutive primes in this context, confirming that 222 is the correct extension of the sequence.

Why Other Options Are Wrong:
Option 168 gives 168 − 1 = 167, which is a prime number and not a product of two primes. Option 196 gives 196 − 1 = 195, which factors as 3*5*13, not a simple product of two consecutive primes. Option 256 gives 256 − 1 = 255, which is 3*5*17, again not matching the pattern. Therefore these values break the rule observed in the known terms.

Common Pitfalls:
Many students start by computing differences between consecutive terms and look for second differences or ratios, which here do not form a simple pattern. Another pitfall is to assume a polynomial rule without checking for factorization patterns. When sequences involve slightly irregular growth, it is often useful to examine n − 1, n + 1 or factorization of nearby values, especially when primes or products are hinted at by the sizes of the numbers.

Final Answer:
The number that should replace the question mark is 222, which corresponds to option C.