Finding the Missing Number in a Series

The question asks us to find the missing number in the given series: 129, 126, 123, 120, 117, ?

To find the missing number, we need to identify the pattern or rule that governs the number series. Let's look at the differences between consecutive terms.

• Difference between the first and second term: \(126 - 129 = -3\)
• Difference between the second and third term: \(123 - 126 = -3\)
• Difference between the third and fourth term: \(120 - 123 = -3\)
• Difference between the fourth and fifth term: \(117 - 120 = -3\)

We observe that the difference between each consecutive term is constant, which is -3. This means the series is an arithmetic progression where each term is obtained by subtracting 3 from the previous term.

To find the next term in the series (the missing number), we need to subtract 3 from the last given term, which is 117.

Calculation:

\(\text{Missing Number} = 117 - 3\)

\(\text{Missing Number} = 114\)

Therefore, the missing number in the series is 114.

Now, let's compare our result with the given options:

• Option 1: 113
• Option 2: 114
• Option 3: 112
• Option 4: 115

Our calculated missing number, 114, matches Option 2.

Revision Table: Number Series Pattern

Additional Information: Arithmetic Series

A number series like the one in this question is called an arithmetic series or arithmetic progression (AP). In an arithmetic series, the difference between consecutive terms is constant. This constant difference is known as the common difference (d).

The general form of an arithmetic series is:

\(a, a+d, a+2d, a+3d, \dots\)

where:

• \(a\) is the first term.
• \(d\) is the common difference.

In the given series 129, 126, 123, 120, 117, ...:

• The first term \(a = 129\).
• The common difference \(d = -3\).

The \(n\)-th term (\(a_n\)) of an arithmetic series can be found using the formula:

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

Let's check the terms using this formula:

• \(a_1 = 129 + (1-1)(-3) = 129 + 0 = 129\) (Correct)
• \(a_2 = 129 + (2-1)(-3) = 129 + (-3) = 126\) (Correct)
• \(a_3 = 129 + (3-1)(-3) = 129 + 2(-3) = 129 - 6 = 123\) (Correct)
• \(a_4 = 129 + (4-1)(-3) = 129 + 3(-3) = 129 - 9 = 120\) (Correct)
• \(a_5 = 129 + (5-1)(-3) = 129 + 4(-3) = 129 - 12 = 117\) (Correct)
• The missing term is the 6th term (\(a_6\)): \(a_6 = 129 + (6-1)(-3) = 129 + 5(-3) = 129 - 15 = 114\) (Correct)

Understanding the concept of arithmetic series helps in solving such missing number and pattern-based reasoning questions.