Understanding the Number Series Question

The question asks us to find the missing number in a given number series: 12, 25, 51, 103, ?, 415, 831. To solve this, we need to identify the pattern or rule that relates consecutive numbers in the series.

Identifying the Pattern in the Series

Let's examine the relationship between the consecutive terms provided in the series:

• From 12 to 25: How do we get from 12 to 25? Let's try simple operations. $12 \times 2 = 24$. If we add 1, we get $24 + 1 = 25$. This matches the second term.
• From 25 to 51: Let's see if the same rule applies. $25 \times 2 = 50$. If we add 1, we get $50 + 1 = 51$. This matches the third term.
• From 51 to 103: Applying the rule again. $51 \times 2 = 102$. Adding 1 gives $102 + 1 = 103$. This matches the fourth term.

It appears the pattern is to multiply the previous number by 2 and then add 1 to get the next number in the series. Let's formulate this pattern:

Next Number = (Previous Number $\times 2$) + 1

Calculating the Missing Number

Now, we can use this pattern to find the number that replaces the question mark (?). The number before the question mark is 103. Applying our identified pattern:

• Previous Number = 103
• Missing Number = (103 $\times 2$) + 1
• Missing Number = $206 + 1$
• Missing Number = 207

So, the missing number is 207.

Verifying the Pattern with Subsequent Terms

Let's check if the number 207 fits correctly in the series by applying the pattern to the subsequent terms.

• From 207 to 415: Using 207 as the previous number. $207 \times 2 = 414$. Adding 1 gives $414 + 1 = 415$. This matches the next term in the series.
• From 415 to 831: Using 415 as the previous number. $415 \times 2 = 830$. Adding 1 gives $830 + 1 = 831$. This matches the last term in the series.

Since the pattern (Previous Number $\times 2 + 1$) holds true for all the given terms, including the missing one, the calculated number 207 is indeed the correct number to replace the question mark.

The complete series is: 12, 25, 51, 103, 207, 415, 831.

Summary of the Pattern

The rule for this number series is that each term is obtained by multiplying the previous term by 2 and adding 1.

Conclusion

Based on the established pattern, the number that replaces the question mark (?) in the series 12, 25, 51, 103, ?, 415, 831 is 207.

Revision Table: Number Series Pattern Analysis

Understanding number series patterns is key to solving such problems. Here's a quick review of the process used:

• Examine the differences or ratios between consecutive terms.
• Look for simple arithmetic operations (addition, subtraction, multiplication, division) or a combination of operations.
• Test the potential pattern across multiple terms in the series.
• Apply the confirmed pattern to find the missing term.
• Verify the result by checking if the pattern continues with the terms following the missing number.

Additional Information: Types of Number Series

Number series questions can follow various patterns. Some common types include:

• Arithmetic Series: A constant difference between consecutive terms (e.g., 2, 4, 6, 8...).
• Geometric Series: A constant ratio between consecutive terms (e.g., 3, 6, 12, 24...).
• Arithmetic-Geometric Series: A combination of arithmetic and geometric progressions.
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5...).
• Difference Series: The differences between consecutive terms form their own pattern (e.g., squared numbers, prime numbers).
• Mixed Series: Involving multiple patterns or operations like the one solved here (multiply by a number and add/subtract another number).

Practicing different types of series helps in quickly identifying the underlying rule during exams.