Number Series — Alternating Operations Using Previous Terms Determine the next number in the sequence: 6, 18, 3, 21, 7, 56, ?

Introduction / Context:
Some sequences alternate between multiplication and division where the divisor can be a previous term. Identifying such interdependencies is important for complex series questions.

Given Data / Assumptions:

• Sequence: 6, 18, 3, 21, 7, 56, ?
• Notice the alternating rise and fall, hinting at multiply then divide patterns.

Concept / Approach:
Test whether a term is obtained by multiplying the immediately previous term by some number, then the next term is obtained by dividing by an earlier term (often the term before the multiplication started). This yields a repeating structural motif.

Step-by-Step Solution:
6 → 18: 6 * 3 = 1818 → 3: 18 / 6 = 3 (divide by the term two places before)3 → 21: 3 * 7 = 2121 → 7: 21 / 3 = 7 (again divide by the term two places before)7 → 56: 7 * 8 = 5656 → ?: divide by the term two places before, i.e., 56 / 7 = 8

Verification / Alternative check:
The pattern is consistent: multiply by a number (3, then 7, then 8), then divide by the term that initiated that mini-pair. Hence, the next term is 8.

Why Other Options Are Wrong:

• 9, 63, 64: These break the established multiply-then-divide-by-previous-term motif.

Common Pitfalls:
Trying to fit a single arithmetic or geometric rule. Look for alternating operations and relationships to earlier terms in the series, not just the immediately preceding one.

Final Answer:
8