Analyzing the Given Number Sequence

The given sequence is 48, 73, 122, 243, 412, 637. We need to find the term that does not follow the pattern established by the other terms in the sequence.

Finding the Pattern: Calculating Differences

Let's calculate the difference between consecutive terms in the sequence:

• Difference 1: $73 - 48 = 25$
• Difference 2: $122 - 73 = 49$
• Difference 3: $243 - 122 = 121$
• Difference 4: $412 - 243 = 169$
• Difference 5: $637 - 412 = 225$

The sequence of differences is 25, 49, 121, 169, 225.

Identifying the Pattern in Differences

Let's look closely at these differences. They appear to be perfect squares:

• $25 = 5^2$
• $49 = 7^2$
• $121 = 11^2$
• $169 = 13^2$
• $225 = 15^2$

The sequence of bases for these squares is 5, 7, 11, 13, 15.

Determining the Rule for the Bases

Let's examine the sequence of bases: 5, 7, 11, 13, 15. What pattern do these numbers follow? These numbers are odd numbers. Let's consider common patterns for such sequences:

• Consecutive odd numbers starting from 5: 5, 7, 9, 11, 13, 15, ... (This pattern includes 9, which is missing in the observed bases between 7 and 11).
• Consecutive prime numbers starting from 5: 5, 7, 11, 13, 17, ... (This pattern matches the first four bases, 5, 7, 11, 13, but the last base observed is 15, not 17).

The pattern of adding squares of consecutive prime numbers starting from 5 seems to fit the sequence for the first five terms. Let's assume this is the intended pattern.

Generating the Sequence Based on the Pattern

If the pattern is adding squares of consecutive prime numbers starting from 5, the differences should be $5^2, 7^2, 11^2, 13^2, 17^2, ...$

• Term 1: 48 (Given)
• Term 2: $48 + 5^2 = 48 + 25 = 73$ (Matches given)
• Term 3: $73 + 7^2 = 73 + 49 = 122$ (Matches given)
• Term 4: $122 + 11^2 = 122 + 121 = 243$ (Matches given)
• Term 5: $243 + 13^2 = 243 + 169 = 412$ (Matches given)
• Term 6: According to the pattern, the next prime after 13 is 17. So, Term 6 should be $412 + 17^2 = 412 + 289 = 701$.

The sequence generated by this pattern is 48, 73, 122, 243, 412, 701.

Identifying the Wrong Term

Comparing the generated sequence with the given sequence:

All terms match the pattern except for the last term. The given last term is 637, but based on the pattern of adding squares of consecutive prime numbers starting from 5, it should be 701.

Therefore, the incorrect term in the given sequence is 637.

Conclusion

The sequence follows the pattern where the difference between consecutive terms is the square of consecutive prime numbers starting from 5. The last term, 637, does not fit this pattern.