Which number will replace the question mark (?) in the following series? 37, ?, 61, 78, 97, 120, 149
48
Solving the Number Series Problem
The question asks us to find the number that replaces the question mark (?) in the given series: 37, ?, 61, 78, 97, 120, 149.
To solve number series problems, we look for a pattern in the sequence. This pattern can be based on addition, subtraction, multiplication, division, squares, cubes, prime numbers, or a combination of these operations applied to the numbers themselves or the differences between consecutive numbers.
Analyzing the Differences Between Consecutive Terms
Let's first find the differences between the known consecutive terms in the series:
- Difference between 149 and 120: $149 - 120 = 29$
- Difference between 120 and 97: $120 - 97 = 23$
- Difference between 97 and 78: $97 - 78 = 19$
- Difference between 78 and 61: $78 - 61 = 17$
So, the differences between the known terms, moving from left to right starting from 61, are: 17, 19, 23, 29.
Identifying the Pattern in the Differences
Let's examine the sequence of differences: 17, 19, 23, 29. We need to identify the pattern in this sequence.
Consider the sequence of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
The differences we found (17, 19, 23, 29) are all prime numbers. Specifically, they are prime numbers in increasing order.
Applying the Pattern to Find the Missing Number
If the pattern is that the differences between consecutive terms are increasing prime numbers, then the differences leading up to 61 must be prime numbers smaller than 17.
Let the missing number be $X$. The differences would be:
- Difference between $X$ and 37: $X - 37$
- Difference between 61 and $X$: $61 - X$
The sequence of differences for the entire series would be: $(X - 37)$, $(61 - X)$, 17, 19, 23, 29.
Following the pattern of increasing prime numbers, the difference $(61 - X)$ must be a prime number smaller than 17. The prime number immediately preceding 17 is 13.
So, let's assume $61 - X = 13$.
Solving for $X$: $X = 61 - 13 = 48$.
Verifying the Pattern
If $X = 48$, the difference between 48 and 37 is $48 - 37 = 11$.
The complete sequence of differences would be: 11, 13, 17, 19, 23, 29.
Let's check if this sequence fits the pattern of increasing prime numbers:
- 11 is a prime number.
- 13 is a prime number.
- 17 is a prime number.
- 19 is a prime number.
- 23 is a prime number.
- 29 is a prime number.
The sequence of differences 11, 13, 17, 19, 23, 29 is a sequence of prime numbers in increasing order. This confirms the pattern.
Therefore, the missing number is 48.
The completed series is: 37, 48, 61, 78, 97, 120, 149.
The differences are: +11, +13, +17, +19, +23, +29.
| Term | Value | Difference from previous term | Pattern |
|---|---|---|---|
| 1st | 37 | - | - |
| 2nd | 48 | $48 - 37 = 11$ | Prime Number |
| 3rd | 61 | $61 - 48 = 13$ | Prime Number |
| 4th | 78 | $78 - 61 = 17$ | Prime Number |
| 5th | 97 | $97 - 78 = 19$ | Prime Number |
| 6th | 120 | $120 - 97 = 23$ | Prime Number |
| 7th | 149 | $149 - 120 = 29$ | Prime Number |
The number that replaces the question mark is 48.
Revision Table - Number Series Patterns
| Pattern Type | Description | Example Series |
|---|---|---|
| Arithmetic Progression | Constant difference between terms. | 2, 5, 8, 11, ... (Difference +3) |
| Geometric Progression | Constant ratio between terms. | 3, 6, 12, 24, ... (Ratio $\times$2) |
| Difference Series | The differences between consecutive terms follow a pattern (e.g., AP, GP, squares, cubes, prime numbers). | Example: The problem solved here, differences are primes. |
| Double Difference Series | The differences between the differences follow a pattern. | 0, 1, 3, 6, 10, ... (Differences: 1, 2, 3, 4; Second differences: 1, 1, 1) |
| Squares/Cubes Series | Terms related to squares or cubes of numbers. | 1, 4, 9, 16, ... ($1^2, 2^2, 3^2, 4^2$) |
| Mixed Series | Combination of two or more patterns. | 1, 5, 2, 10, 3, 15, ... (AP 1, 2, 3... and GP 5, 10, 15...) |
Additional Information - Prime Numbers and Number Series
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, ...
In number series questions, prime numbers are often used in various ways:
- As the terms of the series themselves (e.g., 2, 3, 5, 7, ...).
- As the differences between consecutive terms (as seen in this problem).
- As the numbers being squared or cubed (e.g., $2^2, 3^2, 5^2, ...$).
- In more complex patterns involving arithmetic operations with prime numbers.
Recognizing prime numbers quickly is a useful skill for solving such reasoning problems. When analyzing a number series, especially if the differences don't follow a simple arithmetic or geometric progression, checking if they are prime numbers or related to primes is a common strategy.
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, ?
