Select the number from among the given options that can replace the question mark (?) in the following series. 45, 46, 50, 59, 75, ?
Analyzing the Number Series Pattern
The question asks us to find the next number in the given series: 45, 46, 50, 59, 75, ?
To solve a number series problem, we usually look for a pattern in the differences between consecutive terms, or a pattern in the terms themselves, or a combination of operations applied to each term.
Finding the Differences Between Consecutive Terms
Let's calculate the difference between each term and the previous one:
- Difference between the 2nd term (46) and the 1st term (45): $46 - 45 = 1$
- Difference between the 3rd term (50) and the 2nd term (46): $50 - 46 = 4$
- Difference between the 4th term (59) and the 3rd term (50): $59 - 50 = 9$
- Difference between the 5th term (75) and the 4th term (59): $75 - 59 = 16$
Identifying the Pattern in the Differences
The sequence of differences we found is: 1, 4, 9, 16.
Let's observe this new sequence:
- $1 = 1^2$
- $4 = 2^2$
- $9 = 3^2$
- $16 = 4^2$
The pattern in the differences is that they are consecutive perfect squares: $1^2, 2^2, 3^2, 4^2$.
Predicting the Next Difference
Following this pattern, the next difference in the sequence should be the next perfect square, which is $5^2$.
Next difference $= 5^2 = 25$.
Calculating the Next Term in the Series
To find the next term in the original series (which replaces the question mark), we add the next predicted difference to the last term of the original series.
Last term of the series = 75
Next difference = 25
Next term = Last term + Next difference
Next term $= 75 + 25 = 100$
Summary of the Pattern
| Term Position | Term Value | Difference from Previous Term | Pattern of Difference |
|---|---|---|---|
| 1st | 45 | - | - |
| 2nd | 46 | $46 - 45 = 1$ | $1^2$ |
| 3rd | 50 | $50 - 46 = 4$ | $2^2$ |
| 4th | 59 | $59 - 50 = 9$ | $3^2$ |
| 5th | 75 | $75 - 59 = 16$ | $4^2$ |
| 6th | $75 + 25 = 100$ | $+25$ | $5^2$ |
The next term in the series is 100.
Final Answer Determination
Based on the pattern identified in the differences between consecutive terms, the number that replaces the question mark (?) in the series 45, 46, 50, 59, 75, ? is 100.
Revision Table: Understanding Number Series
| Concept | Description | Example |
|---|---|---|
| Number Series | A sequence of numbers that follow a specific pattern or rule. | 2, 4, 6, 8, ? (Pattern: add 2) |
| Difference Pattern | Analyzing the differences between consecutive terms to find a pattern. | Series: 3, 6, 11, 18. Differences: 3, 5, 7 (Odd numbers) |
| Perfect Squares | Numbers obtained by squaring an integer ($n^2$). | 1, 4, 9, 16, 25, 36, ... |
| Consecutive Terms | Numbers that follow each other immediately in a series. | In 45, 46, 50, 59, 75, 100, 45 and 46 are consecutive. |
Additional Information: Common Number Series Patterns
Number series questions can follow various patterns. Some common types include:
- Arithmetic Progression: The difference between consecutive terms is constant (e.g., 5, 10, 15, 20...).
- Geometric Progression: Each term is multiplied by a constant ratio to get the next term (e.g., 2, 4, 8, 16...).
- Difference Series: The differences between consecutive terms form their own pattern (as seen in this question). This pattern could be arithmetic, geometric, perfect squares, cubes, prime numbers, etc.
- Alternating Series: The pattern alternates between two different rules or operations.
- Fibonacci Series or Similar: Each term is the sum of the previous two terms (e.g., 1, 1, 2, 3, 5, 8...).
- Operations Based: A rule involving multiplication, division, addition, or subtraction applied sequentially or based on the term's position.
Solving number series problems requires careful observation and testing different types of patterns.
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, ?
