Finding the Missing Number in the Series

The question asks us to find the missing number in the given series: 7, 14, 28, 56, ?, 224, 448.

To solve this number series problem, we need to identify the pattern or rule that governs the sequence of numbers.

Analyzing the Number Series Pattern

Let's look at the relationship between consecutive terms in the series:

• The second term is 14, and the first term is 7. $14 \div 7 = 2$.
• The third term is 28, and the second term is 14. $28 \div 14 = 2$.
• The fourth term is 56, and the third term is 28. $56 \div 28 = 2$.

From this analysis, it is clear that each term is obtained by multiplying the previous term by 2. This type of series, where each term after the first is found by multiplying the previous one by a fixed, non-zero number, is called a geometric progression.

The common ratio of this geometric progression is 2.

Step-by-Step Calculation to Find the Missing Number

The series follows the pattern: $a_n = a_{n-1} \times 2$, where $a_n$ is the $n$-th term and $a_{n-1}$ is the previous term.

The series is 7, 14, 28, 56, ?, 224, 448.

We need to find the term after 56. Let the missing term be $x$.

Based on the pattern, the missing term ($x$) should be equal to the previous term (56) multiplied by the common ratio (2).

So, $x = 56 \times 2$.

Calculating the value:

$x = 112$.

Verifying the Pattern with Subsequent Terms

Let's check if the next term in the series (224) follows the pattern using the calculated missing number (112):

$112 \times 2 = 224$. This matches the next number in the given series.

Let's also check the last term (448):

$224 \times 2 = 448$. This matches the last number in the given series.

Thus, the missing number in the series is indeed 112.

Comparing with the Given Options

The given options are:

• 116
• 124
• 112
• 108

Our calculated missing number is 112, which matches one of the options.

Therefore, the option that replaces the question mark (?) is 112.

Revision Table: Number Series Pattern Analysis

Reviewing the key steps involved in solving this number series question:

• Examine the relationship between adjacent numbers in the sequence.
• Look for common differences (arithmetic progression) or common ratios (geometric progression).
• Apply the identified pattern to find the missing term.
• Verify the pattern using subsequent terms if available.

Additional Information: Types of Number Series

Number series questions are common in various exams. Besides geometric progressions, other types include:

• Arithmetic Progression: Each term is obtained by adding a fixed number to the previous term (e.g., 3, 6, 9, 12...).
• Difference Series: The difference between consecutive terms follows a pattern (e.g., differences are increasing, decreasing, or form another simple series).
• Square/Cube Series: Terms are squares or cubes of natural numbers, possibly with some additions or subtractions (e.g., 1, 4, 9, 16...).
• Mixed Series: A combination of two or more patterns, or two interleaved series.
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5, 8...).

Identifying the correct type of pattern is crucial for solving number series problems.