Select the number from among the given options that can replace the question mark (?) in the following series. 20, 25, 35, 50, 70, ?
95
Understanding the Number Series Problem
The question asks us to find the next number in the given series: 20, 25, 35, 50, 70, ?. To solve this number series problem, we need to identify the pattern or rule that governs the sequence of numbers.
Finding the Pattern in the Number Series
A common approach to solving number series is to look at the difference between consecutive terms. Let's calculate the difference between each pair of adjacent numbers in the series:
- Difference between the 2nd and 1st term: 25 - 20 = 5
- Difference between the 3rd and 2nd term: 35 - 25 = 10
- Difference between the 4th and 3rd term: 50 - 35 = 15
- Difference between the 5th and 4th term: 70 - 50 = 20
The differences we found are 5, 10, 15, and 20.
Analyzing the Differences: A Pattern Within a Pattern
Now, let's look closely at the sequence of differences: 5, 10, 15, 20. We can see a clear pattern here. These differences are increasing by a constant amount.
- The difference between 10 and 5 is 10 - 5 = 5.
- The difference between 15 and 10 is 15 - 10 = 5.
- The difference between 20 and 15 is 20 - 15 = 5.
This shows that the differences between the terms of the original series form an arithmetic progression with a common difference of 5. This is a pattern where the difference between consecutive terms itself increases by 5 each time.
Predicting the Next Difference and Term
Following the pattern of the differences (5, 10, 15, 20), the next difference should be the last difference (20) plus 5. So, the next difference is:
\(20 + 5 = 25\)
To find the next number in the original series, we add this predicted difference (25) to the last term of the series, which is 70.
\(70 + 25 = 95\)
Conclusion: The Missing Number
Based on the identified pattern, the number that should replace the question mark is 95.
| Term | Value | Difference from Previous Term |
|---|---|---|
| 1st | 20 | - |
| 2nd | 25 | \(25 - 20 = 5\) |
| 3rd | 35 | \(35 - 25 = 10\) |
| 4th | 50 | \(50 - 35 = 15\) |
| 5th | 70 | \(70 - 50 = 20\) |
| 6th | ? | \(70 + 25 = 95\) |
Revision Table: Key Steps for Number Series
| Step | Description |
|---|---|
| 1 | Calculate differences between consecutive terms. |
| 2 | Look for a pattern in these differences (e.g., arithmetic progression, geometric progression, constant). |
| 3 | If no simple pattern, look at differences of differences (second-order differences). |
| 4 | Predict the next difference based on the pattern found. |
| 5 | Add the predicted difference to the last known term to find the next term. |
Additional Information: Exploring Number Series Types
Number series questions test your ability to find patterns. Besides the pattern seen here (differences forming an arithmetic series), other common types include:
- Arithmetic Series: Each term is obtained by adding a constant value (common difference) to the previous term (e.g., 2, 4, 6, 8...).
- Geometric Series: Each term is obtained by multiplying the previous term by a constant value (common ratio) (e.g., 3, 6, 12, 24...).
- Fibonacci Series: Each term is the sum of the two preceding terms (starting usually from 0 and 1, or 1 and 1) (e.g., 0, 1, 1, 2, 3, 5, 8...).
- Mixed Series: Combination of two or more patterns.
- Prime Number Series: Series composed entirely of prime numbers (e.g., 2, 3, 5, 7, 11...).
- Square/Cube Series: Series involving squares or cubes of numbers (e.g., 1, 4, 9, 16... or 1, 8, 27, 64...).
Practice with different types of series helps in quickly identifying the underlying pattern during exams.
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