In the number series 2, 6, 12, 20, 30, 42, 56, ( ... ), what is the next term that should follow 56?

Introduction / Context:
This question provides a series of numbers that increase in a non linear but regular way. By examining the differences between consecutive terms, we can spot a clear pattern. Such problems are designed to test your ability to detect patterns in sequences and to use first differences to understand how the series grows.

Given Data / Assumptions:

• The series is 2, 6, 12, 20, 30, 42, 56, ( ... ).
• We are asked to find the next term after 56.
• The terms are positive and the increments between them are increasing.

Concept / Approach:
To uncover the pattern, calculate the difference between each pair of successive terms. If these differences themselves follow a simple rule, such as forming an arithmetic progression, we can extend that rule to predict the next difference and hence the next term. In this series, the differences turn out to be consecutive even numbers, which makes extrapolation straightforward.

Step-by-Step Solution:
Compute the differences between consecutive terms: 6 - 2 = 4. 12 - 6 = 6. 20 - 12 = 8. 30 - 20 = 10. 42 - 30 = 12. 56 - 42 = 14. The differences are 4, 6, 8, 10, 12, 14. These differences form an arithmetic progression with first term 4 and common difference 2. Following this pattern, the next difference should be 16. Therefore, the next term after 56 is 56 + 16 = 72. So the required next term is 72.

Verification / Alternative Check:
We can interpret each term as the sum of consecutive even numbers starting from 2. For example, 2 = 2, 6 = 2 + 4, 12 = 2 + 4 + 6, 20 = 2 + 4 + 6 + 8, and so on. Under this interpretation, 56 represents the sum 2 + 4 + 6 + ... + 14. The next term would then be the sum 2 + 4 + 6 + ... + 16, which is 56 + 16 = 72. This alternative view is consistent with the difference pattern and confirms that the correct next term is 72.

Why Other Options Are Wrong:

• Option 61: This would correspond to adding a difference of only 5 to 56, which does not match the even number difference pattern.
• Option 64: This implies a difference of 8 from 56, breaking the steady growth of the differences.
• Option 70: This uses a difference of 14 again instead of increasing it to 16, disrupting the pattern of consecutive even increments.
• Option 68: This adds a difference of 12, which has already been used earlier in the series.

Common Pitfalls:
Some students might initially think the series could be quadratic or some more complex function and overlook the simple pattern in the differences. Others may miscalculate one of the differences, which can hide the clear sequence of 4, 6, 8, 10, 12, 14. It is important to compute differences carefully and then see if those differences themselves form an arithmetic progression.

Final Answer:
The next term in the series after 56 is 72.