Understanding the Number Series Problem

This question asks us to find the missing number in a given series: 1.14, 1.28, 1.42, ?, 1.70, 1.84.

To solve a number series problem, we need to identify the pattern or rule that connects consecutive numbers in the sequence. Once the pattern is found, we can apply it to determine the missing term.

Finding the Pattern in the Series

Let's examine the difference between consecutive terms in the given series:

• Difference between the second term and the first term:
• \(\text{1.28} - \text{1.14} = \text{0.14}\)
• Difference between the third term and the second term:
• \(\text{1.42} - \text{1.28} = \text{0.14}\)
• Difference between the sixth term and the fifth term:
• \(\text{1.84} - \text{1.70} = \text{0.14}\)

It appears that the difference between each consecutive term is constant, which is 0.14. This indicates that the series is an arithmetic progression where each term is obtained by adding 0.14 to the previous term.

Calculating the Missing Number

The missing number is the fourth term in the series. According to the pattern, the fourth term should be the third term plus the common difference.

Missing Term = Third Term + Common Difference

Missing Term = \(\text{1.42} + \text{0.14}\)

Let's perform the addition:

\(\begin{array}{@{}c@{\,}c@{}c@{}c} & 1 & . & 4 & 2 \\ + & 0 & . & 1 & 4 \\ \hline & 1 & . & 5 & 6 \\ \end{array}\)

So, the missing number is 1.56.

Verifying the Pattern

Let's check if the next term in the series follows the pattern using the calculated missing number (1.56).

• Fifth term = Missing Term + Common Difference
• Fifth term = \(\text{1.56} + \text{0.14}\)
• \(\text{1.56} + \text{0.14} = \text{1.70}\)

This matches the given fifth term in the series (1.70). Therefore, the pattern is consistent, and the calculated missing number is correct.

Conclusion

The series follows a pattern where each term is obtained by adding 0.14 to the previous term. By applying this pattern, the missing number is found to be 1.56.

The completed series is: 1.14, 1.28, 1.42, 1.56, 1.70, 1.84.

Number Series Problem Solving Revision

When tackling number series questions, remember these steps:

1. Look for common differences between consecutive terms. This identifies arithmetic series.
2. Look for common ratios between consecutive terms. This identifies geometric series.
3. Check for patterns involving squares, cubes, or prime numbers.
4. Consider alternating patterns or patterns involving differences of differences.
5. Practice solving various types of series to become familiar with common patterns.

Additional Information on Arithmetic Series

An arithmetic series is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by 'd'.

The general form of an arithmetic series is:

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

where 'a' is the first term and 'd' is the common difference.

In the given problem, the first term \(a = 1.14\) and the common difference \(d = 0.14\).

The nth term of an arithmetic series can be found using the formula:

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

For example, the 4th term (missing term) is \(a_4 = 1.14 + (4-1) \times 0.14 = 1.14 + 3 \times 0.14 = 1.14 + 0.42 = 1.56\).