Number series – insert the missing term (constant second difference 8): 3, 15, ?, 63, 99, 143 Find the middle term so that first differences rise by 8 each step.

Introduction / Context:
Another standard device is an arithmetic progression of first differences. If the increments themselves climb steadily by a fixed amount k, the series is consistent with a quadratic pattern and can be repaired by inserting the term that preserves that constant second difference.

Given Data / Assumptions:

• Given terms: 3, 15, ?, 63, 99, 143.
• We look for evenly rising first differences.

Concept / Approach:
Let the first differences be d1, d2, d3, d4, d5 with d(i+1) − d(i) = 8. From the tail (63→99→143), the differences are 36 and 44, consistent with a +8 step. Work backward to place the missing value that preserves the pattern from the front as well.

Step-by-Step Solution:

Verification / Alternative check:
Confirm entire chain of differences: 12, 20, 28, 36, 44 → constant second difference 8.

Why Other Options Are Wrong:

Common Pitfalls:
Guessing “middle” by averaging neighbors; that does not preserve the higher-order difference structure.

Final Answer:
35