Which of the following numbers will replace the question mark (?) in the given series? 40, 53, 68, ?, 104, 125, 148
85
Solving the Number Series: Finding the Missing Term
This question asks us to find the missing number in the given series: 40, 53, 68, ?, 104, 125, 148. To solve this type of number series problem, we need to identify the underlying pattern that relates consecutive terms.
Let's examine the differences between consecutive terms in the given series where numbers are provided:
- Difference between the second and first term: $53 - 40 = 13$
- Difference between the third and second term: $68 - 53 = 15$
- Difference between the sixth and fifth term: $125 - 104 = 21$
- Difference between the seventh and sixth term: $148 - 125 = 23$
Now let's look at the sequence of these differences: 13, 15, ?, 21, 23.
We can observe a pattern in these differences. The differences are increasing by 2 each time:
- $15 - 13 = 2$
- $23 - 21 = 2$
This suggests that the differences form an arithmetic progression with a common difference of 2.
Following this pattern, the difference between the third term (68) and the missing term (?) should be the next term in the difference series after 15. This would be $15 + 2 = 17$.
So, the missing term is $68 + 17$.
Calculating the missing term:
Missing Term $= 68 + 17 = 85$.
Let's verify this by checking the next difference. The difference between the missing term (85) and the fifth term (104) should be the next term in the difference series after 17. This would be $17 + 2 = 19$.
Checking the difference between 104 and 85: $104 - 85 = 19$.
This confirms our pattern. The sequence of differences is 13, 15, 17, 19, 21, 23.
The complete series with the missing number is therefore: 40, 53, 68, 85, 104, 125, 148.
The number that replaces the question mark is 85.
Let's summarize the series and the differences:
| Term Number | Term Value | Difference from Previous Term |
|---|---|---|
| 1 | 40 | - |
| 2 | 53 | $53 - 40 = 13$ |
| 3 | 68 | $68 - 53 = 15$ |
| 4 | 85 | $85 - 68 = 17$ |
| 5 | 104 | $104 - 85 = 19$ |
| 6 | 125 | $125 - 104 = 21$ |
| 7 | 148 | $148 - 125 = 23$ |
The pattern of differences (13, 15, 17, 19, 21, 23) confirms that 85 is the correct missing number.
Revision Table: Number Series Patterns
Understanding common patterns is key to solving number series questions. Here are a few types:
| Pattern Type | Description | Example |
|---|---|---|
| Arithmetic Progression | Adding or subtracting a constant value. | 2, 5, 8, 11, ... (Add 3) |
| Geometric Progression | Multiplying or dividing by a constant value. | 3, 6, 12, 24, ... (Multiply by 2) |
| Difference Series | The differences between terms follow a pattern (like arithmetic, geometric, etc.). (This question's type) | 1, 2, 4, 7, 11, ... (Differences: 1, 2, 3, 4) |
| Double Difference Series | The differences between the differences follow a pattern. | 1, 3, 7, 13, 21, ... (Differences: 2, 4, 6, 8) (Double Differences: 2, 2, 2) |
| Squares or Cubes | Terms related to squares ($n^2$) or cubes ($n^3$) of natural numbers. | 1, 4, 9, 16, ... ($1^2, 2^2, 3^2, 4^2$) |
| Combination Series | A mix of two or more patterns. | 2, 5, 10, 17, ... ($n^2 + 1$) |
Additional Information: Number Series Tips
When tackling number series problems in exams, keep these tips in mind:
- Always check the differences between consecutive terms first. This is a very common pattern type.
- If the first differences don't show a clear pattern, check the differences between the differences (double difference).
- Look for patterns involving multiplication or division, especially if the numbers are growing or shrinking rapidly.
- Consider if the terms are related to squares, cubes, or prime numbers.
- Sometimes, the pattern might involve alternating operations or combining simple sequences.
- Practice with various types of series to become familiar with different patterns.
Similar Questions
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Important Questions from Number Series
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दिए गए विकल्पों में से वह संख्या चुनिए जो निम्नलिखित श्रृंखला में प्रश्नवाचक चिन्ह (?) को प्रतिस्थापित कर सके।
20, 21, 25, 34,?, 75
Select the correct option that will fill in the blank and complete the series.
45, 49, 58, 74, .........
