Which number will replace the question mark (?) in the following series? 27, 32, 42, 62, ?, 182
102
Understanding the Number Series Pattern
The question asks us to find the missing number in the given series: 27, 32, 42, 62, ?, 182. To solve this, we need to identify the pattern or rule that governs the sequence of numbers.
Analyzing the Differences Between Terms
Let's look at the differences between consecutive terms in the series:
- Difference between the 2nd and 1st term: \(32 - 27 = 5\)
- Difference between the 3rd and 2nd term: \(42 - 32 = 10\)
- Difference between the 4th and 3rd term: \(62 - 42 = 20\)
We can observe a pattern in these differences: 5, 10, 20. Each difference is double the previous difference.
Identifying the Pattern in Differences
The pattern in the differences is:
- \(10 = 5 \times 2\)
- \(20 = 10 \times 2\)
Following this pattern, the next difference should be double the last calculated difference (20).
Next difference = \(20 \times 2 = 40\)
Calculating the Missing Number
The missing number (?) is the 5th term in the series. To find the 5th term, we add the next difference (40) to the 4th term (62).
Missing number = 4th term + next difference
Missing number = \(62 + 40 = 102\)
Verifying the Pattern with the Last Term
Now that we have found the missing number (102), let's check if the pattern continues with the last term (182). The difference between the last term (182) and the number we found (102) should follow the established pattern of differences.
Difference between the 6th and 5th term: \(182 - 102 = 80\)
The previous difference was 40. Is 80 double of 40?
\(40 \times 2 = 80\)
Yes, the pattern holds true. The differences are 5, 10, 20, 40, 80, where each difference is double the previous one.
Conclusion of the Number Series Analysis
The series with the missing number filled in is: 27, 32, 42, 62, 102, 182.
The number that replaces the question mark is 102.
| Term Position | Term Value | Difference from Previous Term |
|---|---|---|
| 1st | 27 | - |
| 2nd | 32 | \(32 - 27 = 5\) |
| 3rd | 42 | \(42 - 32 = 10\) |
| 4th | 62 | \(62 - 42 = 20\) |
| 5th (?) | 102 | \(102 - 62 = 40\) |
| 6th | 182 | \(182 - 102 = 80\) |
Revision Table: Number Series Pattern
This table summarizes the pattern found in the differences between consecutive terms:
| Difference | Calculation | Pattern |
|---|---|---|
| First Difference | 5 | Base difference |
| Second Difference | 10 | \(5 \times 2\) |
| Third Difference | 20 | \(10 \times 2\) |
| Fourth Difference | 40 | \(20 \times 2\) |
| Fifth Difference | 80 | \(40 \times 2\) |
Additional Information: Solving Number Series Problems
Solving number series problems often involves looking for patterns in the differences or ratios between consecutive terms. Common patterns include:
- Arithmetic Progression (constant difference)
- Geometric Progression (constant ratio)
- Differences forming an Arithmetic or Geometric Progression
- Alternating patterns
- Patterns based on squares, cubes, or other mathematical operations
- Combinations of patterns
It is helpful to calculate the differences between consecutive terms first, and if a clear pattern doesn't emerge there, calculate the differences of the differences (second-order differences), and so on. Checking ratios can also reveal geometric 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, ?
