Select the number from among the given options that can replace the question mark (?) in the following series. 37, 52, 74, 104, 143, ?
192
Solving the Number Series Pattern
This question asks us to find the next number in the given series: 37, 52, 74, 104, 143, ?
To solve number series problems, we look for a pattern in the differences between consecutive terms, or sometimes in the ratio or other mathematical relationships.
Analyzing the Differences
Let's calculate the difference between each consecutive pair of numbers in the series:
- Difference between the 2nd and 1st term: $52 - 37 = 15$
- Difference between the 3rd and 2nd term: $74 - 52 = 22$
- Difference between the 4th and 3rd term: $104 - 74 = 30$
- Difference between the 5th and 4th term: $143 - 104 = 39$
The sequence of these first differences is: 15, 22, 30, 39.
Finding the Pattern in Differences
Now, let's look for a pattern in these first differences by finding the differences between them (second differences):
- Difference between the 2nd and 1st first difference: $22 - 15 = 7$
- Difference between the 3rd and 2nd first difference: $30 - 22 = 8$
- Difference between the 4th and 3rd first difference: $39 - 30 = 9$
The sequence of second differences is: 7, 8, 9.
We can see a clear pattern here: the second differences are increasing by 1 each time.
Predicting the Next Number in the Series
Following the pattern of the second differences (7, 8, 9), the next second difference should be $9 + 1 = 10$.
This means the next first difference (which is the difference between the 6th and 5th term) should be the last first difference (39) plus the next second difference (10).
Next first difference $= 39 + 10 = 49$.
The next number in the series will be the last number in the series (143) plus this next first difference (49).
Next number $= 143 + 49 = 192$.
Summary of the Pattern
| Term | Value | First Difference | Second Difference |
|---|---|---|---|
| 1st | 37 | - | - |
| 2nd | 52 | $52 - 37 = 15$ | - |
| 3rd | 74 | $74 - 52 = 22$ | $22 - 15 = 7$ |
| 4th | 104 | $104 - 74 = 30$ | $30 - 22 = 8$ |
| 5th | 143 | $143 - 104 = 39$ | $39 - 30 = 9$ |
| 6th | ? | $39 + 10 = 49$ (Predicted) | $9 + 1 = 10$ (Predicted) |
| Result | $143 + 49 = 192$ | - | - |
Conclusion
Following the identified pattern of second differences, the next number in the series is 192.
Revision Table: Key Concepts in Number Series
Understanding different types of patterns is crucial for solving number series problems in competitive exams.
| Pattern Type | Description | Example |
|---|---|---|
| Arithmetic Series | Constant difference between terms. | 2, 5, 8, 11, ... (Difference is 3) |
| Geometric Series | Constant ratio between terms. | 3, 6, 12, 24, ... (Ratio is 2) |
| Difference Series | The differences between terms follow a pattern (like arithmetic, geometric, or another series). This question is an example of a difference series where the second differences form an arithmetic series. | 1, 2, 4, 7, 11, ... (Differences are 1, 2, 3, 4) |
| Mixed Series | Combination of different patterns or operations. | 1, 4, 9, 16, 25, ... (Squares of numbers) |
Additional Information: Tips for Solving Number Series Questions
- Always start by calculating the differences between consecutive terms.
- If the first differences don't show a clear pattern, calculate the second differences, third differences, and so on.
- Look for common patterns like arithmetic progression, geometric progression, squares, cubes, prime numbers, Fibonacci sequence, or alternating patterns.
- Consider multiplication or division patterns if the numbers are growing or shrinking rapidly.
- Practice is key to recognizing different types of number series patterns quickly.
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A series is given with one term wrong. Select that wrong term from the given alternatives.
J12, M24, P48, S96, U192
