Select the number from among the given options that can replace the question mark (?) in the following series. 864, 432, 216, 108,?
54
Understanding the Number Series Pattern
The question asks us to find the next number in the given series: 864, 432, 216, 108, ?. To solve this type of number series problem, we need to identify the pattern or rule that connects the numbers in the sequence.
Let's look at the relationship between consecutive terms in the series:
- From the first term (864) to the second term (432)
- From the second term (432) to the third term (216)
- From the third term (216) to the fourth term (108)
Identifying the Pattern in the Series
We can try simple arithmetic operations like addition, subtraction, multiplication, or division to see if a pattern emerges.
Let's test division:
- $864 \div 2 = 432$
- $432 \div 2 = 216$
- $216 \div 2 = 108$
It is clear from these calculations that each term in the series is obtained by dividing the previous term by 2. This is a consistent pattern of division by 2.
Let's verify this with the series:
| Term 1 | Operation | Term 2 |
|---|---|---|
| 864 | $\div 2$ | 432 |
| Term 2 | Operation | Term 3 |
| 432 | $\div 2$ | 216 |
| Term 3 | Operation | Term 4 |
| 216 | $\div 2$ | 108 |
The pattern is confirmed: divide by 2 to get the next number in the series.
Calculating the Next Number
To find the missing number (?), we apply the same pattern to the last known term in the series, which is 108.
The next number will be $108 \div 2$.
Calculation:
$$108 \div 2 = 54$$So, the next number in the series is 54.
Conclusion
The number that replaces the question mark in the series 864, 432, 216, 108, ? is 54. This is based on the pattern of dividing each term by 2 to get the next term.
Revision Table: Steps to Solve Number Series Problems
| Step | Action | Example (this series) |
|---|---|---|
| 1 | Examine the series and look for relationships between consecutive numbers. | Look at 864 & 432, 432 & 216, etc. |
| 2 | Test common patterns: addition, subtraction, multiplication, division, squares, cubes, differences between terms, etc. | Tried division: $864/2=432$, $432/2=216$, $216/2=108$. Pattern found! |
| 3 | Confirm the pattern holds for all given terms. | Pattern holds consistently. |
| 4 | Apply the identified pattern to the last term to find the missing number. | $108 / 2 = 54$. |
Additional Information on Number Series Patterns
Number series questions can involve various patterns. Some common types include:
- Arithmetic Series: A constant difference between consecutive terms (e.g., 2, 4, 6, 8...).
- Geometric Series: A constant ratio between consecutive terms (like the series in this question, where the ratio is 1/2 or dividing by 2).
- Difference Series: The differences between consecutive terms form a pattern (e.g., the differences might be increasing by a constant, or they might form an arithmetic/geometric series themselves).
- Mixed Series: Combinations of different operations or patterns.
- Fibonacci Series: Each term is the sum of the two preceding ones (e.g., 1, 1, 2, 3, 5, 8...).
Practicing different types of series helps in quickly identifying the pattern during exams.
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