Which of the following numbers will replace the question mark (?) in the given series? 13, 26, 23, 115, 108, ?
1188
Understanding the Number Series Pattern
The question asks us to find the next number in the given series: 13, 26, 23, 115, 108, ?
To solve this number series problem, we need to carefully examine the relationship between consecutive terms and identify the underlying pattern or rule.
Analyzing the Steps Between Terms
Let's look at the operations that transform each term into the next:
- From 13 to 26: $26 \div 13 = 2$. This looks like multiplication by 2.
- From 26 to 23: $23 - 26 = -3$. This looks like subtraction of 3.
- From 23 to 115: $115 \div 23 = 5$. This looks like multiplication by 5.
- From 115 to 108: $108 - 115 = -7$. This looks like subtraction of 7.
Identifying the Pattern Rule
Let's list the operations and the numbers involved:
- Step 1: $13 \xrightarrow{\times 2} 26$
- Step 2: $26 \xrightarrow{- 3} 23$
- Step 3: $23 \xrightarrow{\times 5} 115$
- Step 4: $115 \xrightarrow{- 7} 108$
The operations alternate between multiplication and subtraction. The numbers used in the operations are 2, 3, 5, 7. These are the first four prime numbers in increasing order.
So, the pattern is: Multiply by the 1st prime, subtract the 2nd prime, multiply by the 3rd prime, subtract the 4th prime, and so on, using consecutive prime numbers.
Calculating the Missing Number
Following the identified pattern, the next operation after subtracting the 4th prime (7) should be multiplying by the 5th prime number. The sequence of prime numbers is 2, 3, 5, 7, 11, 13, ...
The 5th prime number is 11.
The last number in the series is 108. We need to apply the next operation (multiplication by 11) to 108.
Calculation:
$\text{Next number} = 108 \times 11$
$108 \times 11 = 108 \times (10 + 1) = (108 \times 10) + (108 \times 1)$
$1080 + 108 = 1188$
Therefore, the number that replaces the question mark is 1188.
| Step | Operation | Prime Number | Calculation | Result |
|---|---|---|---|---|
| 1 | Multiply | 2 (1st prime) | $13 \times 2$ | 26 |
| 2 | Subtract | 3 (2nd prime) | $26 - 3$ | 23 |
| 3 | Multiply | 5 (3rd prime) | $23 \times 5$ | 115 |
| 4 | Subtract | 7 (4th prime) | $115 - 7$ | 108 |
| 5 | Multiply | 11 (5th prime) | $108 \times 11$ | 1188 |
Conclusion
Based on the pattern of alternating multiplication and subtraction using consecutive prime numbers, the next number in the series is 1188.
Revision Table: Number Series Pattern Analysis
| Series Term | Operation to Next Term |
|---|---|
| 13 | $\times 2$ (1st prime) |
| 26 | $- 3$ (2nd prime) |
| 23 | $\times 5$ (3rd prime) |
| 115 | $- 7$ (4th prime) |
| 108 | $\times 11$ (5th prime) |
| 1188 | ... |
Additional Information: Prime Numbers in Number Series
Prime numbers are often used in number series problems to create complex patterns. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The sequence of prime numbers starts with 2, 3, 5, 7, 11, 13, 17, 19, 23, and so on.
In this series, the operations ($\times, -$) alternate, and the magnitude of the operation is the next prime number in sequence. Recognizing the sequence of prime numbers is key to solving such patterns quickly.
Important Questions from Number Series
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Select the number from among the given options that can replace the question mark (?) in the following series.
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