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.
Important Questions from Miscellaneous
A stone is thrown horizontally from the top of a 20 m high building with a speed of 12 m/s. It hits the ground at a distance R from the building. Taking g = 10 m/s2 and neglecting air resistance will give :
A sphere of volume V is made of a material with lower density than water. While on Earth, it floats on water with its volume f1V (f1 < 1) submerged. On the other hand, on a spaceship accelerating with acceleration a < g (g is the acceleration due to gravity on Earth) in outer space, its submerged volume in water is f2V. Then:
A railway wagon (open at the top) of mass M1 is moving with speed v1 along a straight track. As a result of rain, after some time it gets partially filled with water so that the mass of the wagon becomes M2 and speed becomes v2. Taking the rain to be falling vertically and the water stationery inside the wagon, the relation between the two speeds v1 and v2 is :
Consider the following statements:
1. Distance between the longitudes becomes zero on North Pole and South Pole.
2. Distance between the longitudes is maximum on the Equator.
3. Number of longitudes is more than number of latitudes.
Which of the statements given above is/are correct?
One block of 2⋅0 kg mass is placed on top of another block of 3⋅0 kg mass. The coefficient of static friction between the two blocks is 0⋅2. The bottom block is pulled with a horizontal force F such that both the blocks move together without slipping. Taking acceleration due to gravity as 10 m/s2, the maximum value of the frictional force is :
