Select the number from among the given options that can replace the question mark (?) in the following series. 410, 210, 110, 60, 35, ?
22.5
Solving the Number Series: Finding the Missing Term
The question asks us to find the number that replaces the question mark in the given number series: 410, 210, 110, 60, 35, ?
Analyzing the Number Series Pattern
Let's examine the relationship between consecutive terms in the series to identify the underlying pattern. We can look at the differences between adjacent numbers:
- Difference between the 1st and 2nd term: \(410 - 210\)
- Difference between the 2nd and 3rd term: \(210 - 110\)
- Difference between the 3rd and 4th term: \(110 - 60\)
- Difference between the 4th and 5th term: \(60 - 35\)
Identifying the Difference Pattern
Calculating these differences, we get:
- \(410 - 210 = 200\)
- \(210 - 110 = 100\)
- \(110 - 60 = 50\)
- \(60 - 35 = 25\)
The sequence of differences is 200, 100, 50, 25. Let's look for a pattern in this sequence of differences.
- \(200 \div 2 = 100\)
- \(100 \div 2 = 50\)
- \(50 \div 2 = 25\)
It appears that each difference is half of the previous difference. This suggests a consistent pattern in how the terms decrease.
Calculating the Next Term
Following this pattern, the next difference in the series should be half of the last calculated difference (25). Therefore, the next difference is:
\(25 \div 2 = 12.5\)
This next difference (12.5) should be subtracted from the last term in the original series (35) to find the missing term:
\(35 - 12.5\)
Calculating the final value:
\(35 - 12.5 = 22.5\)
So, the missing number in the series is 22.5.
Summary of the Pattern
The pattern involves subtracting a decreasing value from each term to get the next term. The amount subtracted is halved at each step:
- \(410 - 200 = 210\)
- \(210 - 100 = 110\)
- \(110 - 50 = 60\)
- \(60 - 25 = 35\)
- \(35 - 12.5 = 22.5\)
| Term | Value | Difference from Previous Term |
|---|---|---|
| 1st | 410 | - |
| 2nd | 210 | \(410 - 210 = 200\) |
| 3rd | 110 | \(210 - 110 = 100\) |
| 4th | 60 | \(110 - 60 = 50\) |
| 5th | 35 | \(60 - 35 = 25\) |
| 6th | ? | Next Difference: \(25 \div 2 = 12.5\) |
| Missing Term: \(35 - 12.5 = 22.5\) | ||
The number that replaces the question mark is 22.5.
Revision Table: Number Series Concepts
| Concept | Description | Example Pattern Types |
|---|---|---|
| Arithmetic Series | Difference between consecutive terms is constant. | Addition, Subtraction |
| Geometric Series | Ratio between consecutive terms is constant. | Multiplication, Division |
| Difference Series | The differences between consecutive terms follow a pattern (arithmetic, geometric, etc.). | Differences are constant, increasing, decreasing, multiplying, dividing. |
| Mixed Series | Involves a combination of different operations or multiple patterns running simultaneously. | Alternating addition/subtraction, patterns in blocks of terms. |
Additional Information: Solving Number Series Questions
Solving number series problems often requires observing the relationships between numbers. Here are some common strategies:
- Check Differences: Calculate the difference between consecutive terms. Look for a pattern in these differences.
- Check Ratios: For increasing or decreasing series, check the ratio between consecutive terms (term2 / term1).
- Look for Squares/Cubes: Terms might be squares, cubes, or related to squares/cubes (e.g., \(n^2\), \(n^2+1\), \(n^3-1\)).
- Alternating Patterns: The series might have two different patterns applied alternately to terms.
- Combination of Operations: The pattern might involve a combination of operations, like multiply by a number and then add/subtract another number.
- Previous Terms Relationship: A term might be calculated based on the sum, difference, or other operations of the previous one or two terms (like Fibonacci series).
Practice with various types of number series helps in quickly identifying the pattern.
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, ?
