Finding the Missing Number in a Series

Let's analyze the given number series: 25, ?, 43, 67, 115, 211.

We need to identify the pattern that connects the numbers in the series to find the missing term represented by the question mark (?).

Let's look at the differences between consecutive known terms:

• Difference between 67 and 43: $67 - 43 = 24$
• Difference between 115 and 67: $115 - 67 = 48$
• Difference between 211 and 115: $211 - 115 = 96$

The differences we found are 24, 48, and 96. Let's examine the relationship between these differences:

• $48 = 24 \times 2$
• $96 = 48 \times 2$

It appears that each difference is double the previous difference. This suggests a pattern where the differences between consecutive terms are doubling.

Let the difference between 43 and the missing number (?) be $d_2$, and the difference between the missing number (?) and 25 be $d_1$. Following the doubling pattern, we expect:

• $24 = d_2 \times 2$ (if reading forward from ?)
• $d_2 = d_1 \times 2$

Working backward from the known differences (24, 48, 96), the difference before 24 should be $24 / 2 = 12$. This would be the difference between 43 and the missing number.

So, $43 - ? = 12$.

Solving for ?: $? = 43 - 12 = 31$.

Now let's check if the pattern holds for the term before 31 (which is 25). The difference before 12 should be $12 / 2 = 6$. This should be the difference between 31 and 25.

Let's check: $31 - 25 = 6$. This matches the expected difference.

So, the sequence of differences is 6, 12, 24, 48, 96, where each difference is double the previous one.

The series with the missing number is:

• $25 \xrightarrow{+6} 31$
• $31 \xrightarrow{+12} 43$
• $43 \xrightarrow{+24} 67$
• $67 \xrightarrow{+48} 115$
• $115 \xrightarrow{+96} 211$

The pattern holds true. The missing number is 31.

The differences are 6, 12, 24, 48, 96, which is a geometric progression with a common ratio of 2.

Therefore, the number that replaces the question mark is 31.

Revision Table: Number Series Analysis

Additional Information: Solving Number Series Problems

Solving number series questions often involves looking for patterns in the relationship between consecutive numbers. Here are some common strategies:

• Check the difference: Calculate the difference between consecutive terms. See if these differences form a recognisable pattern (e.g., constant, arithmetic progression, geometric progression, squares, cubes).
• Check the ratio: If the numbers are growing or shrinking rapidly, look at the ratio between consecutive terms. This is useful for geometric series.
• Look for patterns in alternate terms: Sometimes, the pattern applies not to consecutive terms but to every other term.
• Check for squares, cubes, or other powers: The terms might be related to squares ($1, 4, 9, 16, ...$), cubes ($1, 8, 27, 64, ...$), or powers of a base number ($2, 4, 8, 16, ...$).
• Combine operations: The pattern might involve a combination of operations, like multiplying by a number and then adding or subtracting another number ($\times 2 + 1$, $\times 3 - 2$, etc.).
• Fibonacci or similar sequences: Check if a term is the sum or difference of the previous two terms, or some other linear combination.
• Position-based patterns: The term might be related to its position in the series (e.g., $n^2$, $n^3$, $2n-1$, etc., where $n$ is the term number).

In this particular series, the pattern was found by examining the differences between terms, which formed a geometric sequence (doubling). This highlights the importance of checking the differences as a first step in number series problems.