Which of the following numbers will replace the question mark (?) in the given series? 18, 23, ?, 41, 54, 71
30
Finding the Missing Number in the Given Series
The problem asks us to identify the number that correctly replaces the question mark (?) in the series: 18, 23, ?, 41, 54, 71.
To solve number series problems like this, we typically look for a pattern or a rule that connects consecutive terms in the series. This pattern might involve addition, subtraction, multiplication, division, squares, cubes, or a combination of these operations. Often, examining the difference between consecutive terms helps reveal the pattern.
Analyzing the Series Pattern
Let's calculate the differences between the consecutive terms we know:
- Difference between the 2nd and 1st term: $23 - 18 = 5$
- Difference between the 5th and 4th term: $54 - 41 = 13$
- Difference between the 6th and 5th term: $71 - 54 = 17$
So the sequence of differences looks like: 5, (difference between ? and 23), (difference between 41 and ?), 13, 17.
Let the missing number be $x$. The differences are: $5$, $x - 23$, $41 - x$, $13$, $17$.
We need to find a value for $x$ from the given options (29, 30, 32, 34) such that these differences follow a discernible pattern.
Testing the Options
Let's substitute each option for $x$ and check the resulting differences and their patterns.
Option 1: If the missing number is 29
Series: 18, 23, 29, 41, 54, 71
Differences between consecutive terms:
- $23 - 18 = 5$
- $29 - 23 = 6$
- $41 - 29 = 12$
- $54 - 41 = 13$
- $71 - 54 = 17$
Sequence of differences: 5, 6, 12, 13, 17
Let's look at the differences between these differences (second differences):
- $6 - 5 = 1$
- $12 - 6 = 6$
- $13 - 12 = 1$
- $17 - 13 = 4$
Sequence of second differences: 1, 6, 1, 4. This pattern is not very consistent.
Option 2: If the missing number is 30
Series: 18, 23, 30, 41, 54, 71
Differences between consecutive terms:
- $23 - 18 = 5$
- $30 - 23 = 7$
- $41 - 30 = 11$
- $54 - 41 = 13$
- $71 - 54 = 17$
Sequence of differences: 5, 7, 11, 13, 17
Let's look at the differences between these differences (second differences):
- $7 - 5 = 2$
- $11 - 7 = 4$
- $13 - 11 = 2$
- $17 - 13 = 4$
Sequence of second differences: 2, 4, 2, 4. This shows a clear repeating pattern (2, 4).
Option 3: If the missing number is 32
Series: 18, 23, 32, 41, 54, 71
Differences between consecutive terms:
- $23 - 18 = 5$
- $32 - 23 = 9$
- $41 - 32 = 9$
- $54 - 41 = 13$
- $71 - 54 = 17$
Sequence of differences: 5, 9, 9, 13, 17
Second differences: 4, 0, 4, 4. This pattern is less consistent than 2, 4, 2, 4.
Option 4: If the missing number is 34
Series: 18, 23, 34, 41, 54, 71
Differences between consecutive terms:
- $23 - 18 = 5$
- $34 - 23 = 11$
- $41 - 34 = 7$
- $54 - 41 = 13$
- $71 - 54 = 17$
Sequence of differences: 5, 11, 7, 13, 17
Second differences: 6, -4, 6, 4. This does not show a clear pattern.
Conclusion
When the missing number is 30, the differences between consecutive terms are 5, 7, 11, 13, 17. The differences between these differences (second differences) form the repeating pattern 2, 4, 2, 4. This is the most consistent pattern among the options tested.
Therefore, the number that replaces the question mark is 30.
| Term | Value | Difference from Previous Term | Second Difference |
|---|---|---|---|
| 1st | 18 | - | - |
| 2nd | 23 | $23 - 18 = 5$ | - |
| 3rd | 30 | $30 - 23 = 7$ | $7 - 5 = 2$ |
| 4th | 41 | $41 - 30 = 11$ | $11 - 7 = 4$ |
| 5th | 54 | $54 - 41 = 13$ | $13 - 11 = 2$ |
| 6th | 71 | $71 - 54 = 17$ | $17 - 13 = 4$ |
Revision Table: Analyzing Number Series
| Step | Description | Why it's important for Number Series |
|---|---|---|
| 1 | Examine the numbers in the series. | Get an overview of the trend (increasing, decreasing, alternating). |
| 2 | Calculate differences between consecutive terms. | Often reveals a pattern directly or leads to a pattern in subsequent differences. |
| 3 | Calculate second or third differences if needed. | For more complex series, the pattern might appear at a deeper level of differences. |
| 4 | Look for patterns (arithmetic, geometric, squares, cubes, prime numbers, repeating sequences). | Identify the rule governing the series. |
| 5 | Test the pattern to predict the missing or next term. | Confirm if the discovered rule holds true for the known terms and helps find the unknown. |
| 6 | Consider options if multiple choice. | Substitute options to see which one fits a consistent pattern. |
Additional Information: Types of Number Series Patterns
Number series questions can involve various types of patterns. Recognizing these can help solve problems faster.
- Arithmetic Series: A constant difference between terms (e.g., 2, 5, 8, 11... difference is 3).
- Geometric Series: A constant ratio between terms (e.g., 3, 6, 12, 24... ratio is 2).
- Difference Series: The differences between terms form their own pattern (as seen in this question, where the second differences follow a pattern).
- Combined Series: Involves more than one operation or interweaves two different series.
- Prime Number Series: Series consists of prime numbers (e.g., 2, 3, 5, 7, 11...).
- Square/Cube Series: Terms are squares or cubes of numbers, or related to them.
- Alternating Series: The pattern alternates between two different operations or two interleaved sub-series.
- Fibonacci-like Series: Each term is the sum of the two preceding terms (e.g., 1, 1, 2, 3, 5, 8...).
Solving number series problems requires observation, pattern recognition, and testing potential rules.
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, ?
