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
Understanding the Number Series Problem
The problem asks us to find the next number in a given series: 410, 210, 110, 60, 35, ?. To solve this type of number series question, we need to identify the pattern or rule that connects the numbers in the sequence.
Let's examine the differences between consecutive terms in the series to see if there's a consistent pattern.
Finding the Pattern in the Series
We will calculate the difference between each adjacent pair of numbers:
- Difference between the 1st and 2nd term: $410 - 210 = 200$
- Difference between the 2nd and 3rd term: $210 - 110 = 100$
- Difference between the 3rd and 4th term: $110 - 60 = 50$
- Difference between the 4th and 5th term: $60 - 35 = 25$
Let's list the differences we found:
| Terms | Difference |
| 410 to 210 | 200 |
| 210 to 110 | 100 |
| 110 to 60 | 50 |
| 60 to 35 | 25 |
Now, let's look at the sequence of these differences: 200, 100, 50, 25. Do you see a pattern here?
It appears that each difference is half of the previous difference.
- $100 = 200 / 2$
- $50 = 100 / 2$
- $25 = 50 / 2$
This confirms a clear pattern in the differences between terms.
Calculating the Next Term
Following the pattern of the differences, the next difference should be half of the last difference (25).
- Next difference $= 25 / 2 = 12.5$
To find the next term in the original series, we subtract this next difference (12.5) from the last term in the series (35).
- Next term $= 35 - 12.5 = 22.5$
Verifying the Solution
The calculated next term is 22.5. We compare this with the given options:
- Option 1: 23
- Option 2: 25
- Option 3: 20
- Option 4: 22.5
The calculated value, 22.5, matches Option 4.
Conclusion
The pattern in the series 410, 210, 110, 60, 35, ? is that the difference between consecutive terms is halved at each step. By extending this pattern, we found the next term to be 22.5.
Revision Table: Number Series Pattern
| Term n | Value | Difference (Term n - Term n+1) | Difference Pattern |
| 1 | 410 | $410 - 210 = 200$ | $200$ |
| 2 | 210 | $210 - 110 = 100$ | $200 / 2 = 100$ |
| 3 | 110 | $110 - 60 = 50$ | $100 / 2 = 50$ |
| 4 | 60 | $60 - 35 = 25$ | $50 / 2 = 25$ |
| 5 | 35 | $35 - 22.5 = 12.5$ | $25 / 2 = 12.5$ |
| 6 | 22.5 | - | - |
Additional Information on Number Series
Number series questions are common in logical reasoning and quantitative aptitude tests. They require you to identify a pattern or rule that generates the sequence of numbers. Common types of patterns include:
- Arithmetic Series: A constant difference between consecutive terms.
- Geometric Series: A constant ratio between consecutive terms.
- Difference Series: The differences between terms form a separate pattern (as seen in this problem).
- Double Difference Series: The differences of the differences form a pattern.
- Squares or Cubes: The terms are related to squares or cubes of natural numbers.
- Fibonacci Series: Each term is the sum of the two preceding terms.
- Mixed Series: A combination of two or more patterns.
Practicing various types of series helps in quickly identifying the underlying rule and solving the problem efficiently.
Important Questions from Number Series
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215, 231, 256, 292, ?
Select the number from among the given options that can replace the question mark (?) in the following series.
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