Select the number from among the given options that can replace the question mark (?) in the following series. 432, 216, 72, 36, 12, ?
6
Understanding the Number Series Pattern
The question asks us to find the missing number in the given series: 432, 216, 72, 36, 12, ?
To solve this number series problem, we need to identify the mathematical relationship or pattern between consecutive terms.
Identifying the Pattern in the Number Series
Let's examine the relationship between adjacent numbers in the series:
- From 432 to 216: $216 = 432 \div 2$
- From 216 to 72: $72 = 216 \div 3$
- From 72 to 36: $36 = 72 \div 2$
- From 36 to 12: $12 = 36 \div 3$
We can observe a repeating pattern of division by 2 and then division by 3.
The sequence of operations is: $\div 2, \div 3, \div 2, \div 3, \dots$
Calculating the Next Term in the Series
Following the identified pattern, the next operation after dividing by 3 should be dividing by 2.
The last term in the given series is 12. Applying the next operation in the pattern:
$? = 12 \div 2$
$? = 6$
Thus, the next number in the series is 6.
Step-by-Step Calculation Summary
| Term | Operation to get the next term | Calculation |
|---|---|---|
| 432 | $\div 2$ | $432 \div 2 = 216$ |
| 216 | $\div 3$ | $216 \div 3 = 72$ |
| 72 | $\div 2$ | $72 \div 2 = 36$ |
| 36 | $\div 3$ | $36 \div 3 = 12$ |
| 12 | $\div 2$ | $12 \div 2 = 6$ |
Conclusion
Based on the established pattern of alternating division by 2 and division by 3, the number that replaces the question mark is 6.
Revision Table: Number Series Analysis
| Concept | Description | Relevance to Problem |
|---|---|---|
| Number Series | A sequence of numbers that follows a specific pattern or rule. | The problem is to find the missing term in a number series. |
| Pattern Identification | Discovering the mathematical operation(s) linking consecutive terms (e.g., addition, subtraction, multiplication, division, squares, cubes, etc.). | Crucial step to determine how each term is derived from the previous one. |
| Alternating Pattern | A pattern where the rule applied changes with each step, often cycling through a set of operations. | The series uses an alternating $\div 2$ and $\div 3$ pattern. |
Additional Information: Types of Number Series Patterns
Number series problems can have various patterns. Understanding common types can help in solving such questions:
- Arithmetic Series: Each term is obtained by adding a constant difference to the previous term (e.g., 2, 5, 8, 11...).
- Geometric Series: Each term is obtained by multiplying the previous term by a constant ratio (e.g., 3, 6, 12, 24...).
- Difference Series: The pattern is in the difference between consecutive terms (e.g., the differences might form an arithmetic or geometric series).
- Ratio Series: The pattern is in the ratio between consecutive terms, similar to geometric series but the ratio itself might follow a pattern.
- Mixed Series: Combinations of operations (like the example here, alternating division).
- Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 1, 1, 2, 3, 5, 8...).
- Square/Cube Series: Terms might be squares or cubes, or involve operations on squares/cubes.
Analyzing the differences, ratios, or applying common arithmetic operations systematically helps in identifying the specific pattern in a given number series.
Important Questions from Number Series
Select the number from among the given options that can replace the question mark (?) in the following series.
37, 52, 74, 104, 143, ?
Select the number that can replace the question mark (?) in the following series.
17, 19, 22, 27, 34, 45, 58,?Select the number from among the given options that can replace the question mark (?) in the following series.
10, 14, 31, 35, 73, 77, ?
Select the number from among the given options that can replace the question mark (?) in the following series.
215, 231, 256, 292, ?
Select the number from among the given options that can replace the question mark (?) in the following series.
6, 6, 8, 24, 28, 140, ?
