Analyzing the Number Series Pattern

The question asks us to find the next number in the given series: 74, 101, 128, 155, ?.

To solve this type of number series question, we need to identify the pattern or rule that connects consecutive terms.

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

• Difference between the second term (101) and the first term (74): \(101 - 74 = 27\)
• Difference between the third term (128) and the second term (101): \(128 - 101 = 27\)
• Difference between the fourth term (155) and the third term (128): \(155 - 128 = 27\)

We observe that the difference between each consecutive pair of numbers is constant and equal to 27. This indicates that the series is an arithmetic progression where each term is obtained by adding 27 to the previous term.

Calculating the Next Term in the Series

Since the common difference is 27, the next term in the series will be found by adding 27 to the last given term, which is 155.

Next term = Last term + Common difference

Next term = \(155 + 27\)

Let's perform the addition:

So, the next number in the series 74, 101, 128, 155, ? is 182.

Understanding Arithmetic Progressions

This number series is an example of an arithmetic progression (AP). An arithmetic progression is a sequence of numbers such that the difference between the consecutive terms is constant. This constant difference is called the common difference, denoted by \(d\).

In this series:

• First term (\(a_1\)) = 74
• Common difference (\(d\)) = 27

The general form of an arithmetic progression is \(a_1, a_1+d, a_1+2d, a_1+3d, \dots\)

The \(n\)-th term of an AP is given by the formula:

\(a_n = a_1 + (n-1)d\)

In our series, let's check the terms using this formula:

• \(a_1 = 74\)
• \(a_2 = 74 + (2-1) \times 27 = 74 + 1 \times 27 = 74 + 27 = 101\)
• \(a_3 = 74 + (3-1) \times 27 = 74 + 2 \times 27 = 74 + 54 = 128\)
• \(a_4 = 74 + (4-1) \times 27 = 74 + 3 \times 27 = 74 + 81 = 155\)

To find the fifth term (\(a_5\)), which is the question mark, we use the formula:

\(a_5 = a_1 + (5-1)d = 74 + 4 \times 27 = 74 + 108 = 182\)

Both methods (finding the next term by adding the common difference or using the general formula) give the same result.

Step-by-Step Solution Summary

1. Identify the given series: 74, 101, 128, 155, ?.
2. Calculate the difference between consecutive terms to find the pattern.
3. Observe that the difference is constant (\(101 - 74 = 27\), \(128 - 101 = 27\), \(155 - 128 = 27\)).
4. Conclude that the series is an arithmetic progression with a common difference of 27.
5. Add the common difference (27) to the last term (155) to find the next term.
6. \(155 + 27 = 182\).
7. The number that replaces the question mark is 182.

Revision Table: Number Series Analysis

Additional Information: Types of Number Series

Number series questions often involve different patterns. Some common types include:

• Arithmetic Progression (AP): A series where the difference between consecutive terms is constant (common difference). Example: 2, 5, 8, 11, ... (common difference = 3)
• Geometric Progression (GP): A series where the ratio of consecutive terms is constant (common ratio). Example: 3, 6, 12, 24, ... (common ratio = 2)
• Fibonacci Series: A series where each term is the sum of the two preceding terms. Example: 0, 1, 1, 2, 3, 5, 8, ...
• Difference Series: The differences between consecutive terms follow a pattern (e.g., they form an AP or GP themselves).
• Mixed Series: A combination of different patterns or operations.

To solve number series questions, it's important to look for common differences, common ratios, squares, cubes, alternating patterns, or combinations of operations.