Select the number that can replace the question mark (?) in the following series.
75
Solving the Number Series Pattern
The question asks us to find the missing number in the series: 17, 19, 22, 27, 34, 45, 58, ?
To solve this type of number series question, we need to identify the pattern or rule that connects the numbers in the sequence. Let's look at the difference between consecutive terms.
Analysing the Differences in the Number Series
Let's calculate the difference between each number and the one preceding it:
- Difference between 19 and 17: \(19 - 17 = 2\)
- Difference between 22 and 19: \(22 - 19 = 3\)
- Difference between 27 and 22: \(27 - 22 = 5\)
- Difference between 34 and 27: \(34 - 27 = 7\)
- Difference between 45 and 34: \(45 - 34 = 11\)
- Difference between 58 and 45: \(58 - 45 = 13\)
The sequence of differences we found is: 2, 3, 5, 7, 11, 13.
Identifying the Pattern in the Differences
Now, let's examine the sequence of differences: 2, 3, 5, 7, 11, 13. Do you recognise this sequence?
These numbers are prime numbers in increasing order. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
The sequence of prime numbers starts with 2, 3, 5, 7, 11, 13, 17, 19, 23, and so on.
The differences in our number series follow this sequence of prime numbers.
Predicting the Next Difference
Since the differences are consecutive prime numbers, the next difference in the series should be the prime number that comes after 13. The next prime number after 13 is 17.
Calculating the Next Number in the Series
To find the missing number (?), we add the next difference (17) to the last number in the series (58).
Missing number \( = 58 + 17 \)
Let's calculate:
\(58 + 17 = 75\)
Verification
The series with the calculated next number is: 17, 19, 22, 27, 34, 45, 58, 75.
The differences are:
- 19 - 17 = 2 (Prime)
- 22 - 19 = 3 (Prime)
- 27 - 22 = 5 (Prime)
- 34 - 27 = 7 (Prime)
- 45 - 34 = 11 (Prime)
- 58 - 45 = 13 (Prime)
- 75 - 58 = 17 (Prime)
The pattern of adding consecutive prime numbers holds true for the entire series.
Conclusion
The number that replaces the question mark is 75.
| Term | Number | Difference from previous term | Observation |
|---|---|---|---|
| 1st | 17 | - | Starting Term |
| 2nd | 19 | \(19 - 17 = 2\) | 1st Prime Number |
| 3rd | 22 | \(22 - 19 = 3\) | 2nd Prime Number |
| 4th | 27 | \(27 - 22 = 5\) | 3rd Prime Number |
| 5th | 34 | \(34 - 27 = 7\) | 4th Prime Number |
| 6th | 45 | \(45 - 34 = 11\) | 5th Prime Number |
| 7th | 58 | \(58 - 45 = 13\) | 6th Prime Number |
| 8th | ? | \(58 + 17 = 75\) | 7th Prime Number after 13 is 17 |
Revision Table: Number Series and Patterns
| Concept | Description | Example |
|---|---|---|
| Number Series | A sequence of numbers that follows a specific rule or pattern. | 2, 4, 6, 8, ... (adding 2 each time) |
| Arithmetic Series | A series where the difference between consecutive terms is constant. | 5, 10, 15, 20, ... (common difference is 5) |
| Geometric Series | A series where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. | 3, 9, 27, 81, ... (common ratio is 3) |
| Difference Series | Finding the pattern by looking at the differences between consecutive terms. The differences might form another recognizable series (arithmetic, geometric, prime numbers, etc.). | 1, 2, 4, 7, 11, ... Differences: 1, 2, 3, 4, ... (arithmetic difference) |
| Prime Numbers | Natural numbers greater than 1 that cannot be formed by multiplying two smaller natural numbers. | 2, 3, 5, 7, 11, 13, 17, 19, 23, ... |
Additional Information: Solving Number Series Questions
Solving number series problems often involves looking for patterns based on:
- Differences: As seen in this problem, the difference between terms can follow a pattern (constant, arithmetic, geometric, prime, etc.).
- Ratios: For geometric series, there's a common ratio.
- Squares or Cubes: Terms might be related to squares or cubes of sequential numbers.
- Combinations: The pattern might involve a combination of operations (e.g., multiply by a number and then add/subtract another).
- Alternating Patterns: Sometimes, there are two different patterns within the same series, applied to alternate terms.
- Fibonacci-like Sequences: Each term is the sum of the two preceding terms.
It's helpful to practice different types of series to become familiar with common patterns. Always start by calculating the differences or ratios between consecutive terms to see if a simple pattern emerges.
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 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, ?
Select the number from among the given options that can replace the question mark (?) in the following series.
35, 54, 77, 106, 137, ?
