In the series 64, 2, 32, 72, 3, 24, 76, 4, 19, ?, 5, 15.6, each group of three numbers follows a consistent rule. Which number should replace the question mark to preserve the pattern?

Introduction / Context:
This sequence alternates between larger and smaller numbers and includes a decimal entry at the end, which suggests that the series may be organised in blocks rather than as a simple one dimensional progression. Grouping the terms into triplets and checking relationships within each group is a powerful approach for this sort of question.

Given Data / Assumptions:

• The full sequence is: 64, 2, 32, 72, 3, 24, 76, 4, 19, ?, 5, 15.6.
• There is a single missing value between 19 and 5.
• Examining the numbers suggests grouping in triplets: (64, 2, 32), (72, 3, 24), (76, 4, 19), (?, 5, 15.6).
• We assume each triplet follows a similar numeric rule.

Concept / Approach:
Within each group of three, the middle number often acts as an operator or divisor linking the first and third numbers. A very natural idea is to test whether the first number divided by the second equals the third number. If this holds for several triplets, we can assume the same relation should hold for the triplet that contains the missing term and solve for that unknown value using simple algebraic rearrangement.

Step-by-Step Solution:
Step 1: Look at the first triplet: 64, 2, 32. Check the relation 64 ÷ 2 = 32, which is true. Step 2: Look at the second triplet: 72, 3, 24. Check 72 ÷ 3 = 24, which is also true. Step 3: Look at the third triplet: 76, 4, 19. Check 76 ÷ 4 = 19, again true. Step 4: Conclude that in each triplet, first number ÷ second number = third number. Step 5: Apply this rule to the last triplet (?, 5, 15.6). We must have ? ÷ 5 = 15.6. Step 6: Multiply both sides by 5 to solve for the missing term: ? = 15.6 * 5. Step 7: Compute 15.6 * 5 = 78.0, so the missing number is 78.

Verification / Alternative check:
We can verify numerically by reconstructing the entire set of triplets with the proposed value. The four groups become (64, 2, 32), (72, 3, 24), (76, 4, 19), and (78, 5, 15.6). Each satisfies the rule first ÷ second = third. Furthermore, the first terms within the groups form a mild progression (64, 72, 76, 78), and the middle terms rise by 1 each time (2, 3, 4, 5), which is a natural secondary structure supporting our conclusion.

Why Other Options Are Wrong:
If we tested 80, 72 or 70, then first ÷ second would not equal 15.6. For example, 80 ÷ 5 = 16, not 15.6; 72 ÷ 5 = 14.4; 70 ÷ 5 = 14. None of these match the third element of the last triplet. Therefore, they break the consistent rule that held in the previous groups.

Common Pitfalls:
A common mistake is to treat the entire sequence as a single chain and look for one step differences or ratios between every pair of successive numbers. The presence of the repeating small numbers 2, 3, 4, 5 hints that the series should instead be broken into segments. Ignoring this structure makes the pattern appear much more complex than it really is.

Final Answer:
To maintain the relation first ÷ second = third within each triplet, the missing term must be 78.