Composite rule: 12, 15, 36, ?, 480, 2415, 14508 Find the missing fourth term given a pattern “multiply by n, then add 3n”.

Introduction / Context:
Some sequences use an index-based operation that changes each step (e.g., multiply by 1, then add 3; multiply by 2, then add 6; and so on). Recognizing the dependence on the step number n is key.

Given Data / Assumptions:

• Terms (t1 to t7): 12, 15, 36, t4, 480, 2415, 14508.
• We suspect an n-based rule: t_{k+1} = t_k * k + 3k for k = 1, 2, 3, …

Concept / Approach:
Validate the rule on early transitions; if they fit, apply it to find t4 and confirm that subsequent given terms also match under k = 4, 5, 6.

Step-by-Step Solution:
k = 1: 12*1 + 3 = 15 (fits).k = 2: 15*2 + 6 = 36 (fits).k = 3: 36*3 + 9 = 117 ⇒ t4 = 117.k = 4: 117*4 + 12 = 480 (fits).k = 5: 480*5 + 15 = 2415 (fits); k = 6: 2415*6 + 18 = 14508 (fits).

Verification / Alternative check:
The rule produces all later terms exactly, uniquely pinpointing 117 as the only valid t4.

Why Other Options Are Wrong:
115, 109, 121 fail to yield 480 under the k = 4 step, breaking the subsequent chain.

Common Pitfalls:
Assuming a constant multiplier/addend and missing the step-index dependence; not checking consistency beyond the missing term.

Final Answer:
117