Solving the Missing Number in the Series

The problem asks us to find the missing number in the given series: 14, 21, 35, 56, 84, ?

To find the missing number in a series, we need to identify the underlying pattern or rule that connects the terms. Let's examine the differences between consecutive terms in the series.

Analyzing the Differences in the Number Series

We calculate the difference between each term and the previous term:

• Difference between the 2nd and 1st term: \(\text{21} - \text{14} = \text{7}\)
• Difference between the 3rd and 2nd term: \(\text{35} - \text{21} = \text{14}\)
• Difference between the 4th and 3rd term: \(\text{56} - \text{35} = \text{21}\)
• Difference between the 5th and 4th term: \(\text{84} - \text{56} = \text{28}\)

The sequence of differences we found is 7, 14, 21, 28. Let's look at this new sequence to see if there's a pattern.

Identifying the Pattern in Differences

The sequence of differences is 7, 14, 21, 28. We can see that each term in this difference sequence is 7 more than the previous term. This means the differences themselves form an arithmetic progression with a common difference of 7.

• \(\text{14} - \text{7} = \text{7}\)
• \(\text{21} - \text{14} = \text{7}\)
• \(\text{28} - \text{21} = \text{7}\)

This confirms that the common difference of the difference sequence is 7.

Finding the Next Difference

Since the differences follow a pattern of increasing by 7 each time, the next difference after 28 should be:

Next difference = Last difference + 7

Next difference = \(\text{28} + \text{7} = \text{35}\)

Calculating the Missing Number

The missing number in the original series is found by adding the next difference (35) to the last term of the original series (84).

Missing Number = Last term + Next difference

Missing Number = \(\text{84} + \text{35} = \text{119}\)

So, the missing number in the series is 119.

Let's summarize the series and differences in a table:

Adding the next difference (35) to the 5th term (84) gives the 6th term: \(84 + 35 = 119\).

The completed series is 14, 21, 35, 56, 84, 119.

Conclusion

The pattern involves a second level of differences, which form an arithmetic progression. By extending this pattern, we found the missing number.

Revision Table: Number Series Patterns

Additional Information on Number Series Reasoning

Number series questions are common in reasoning and quantitative aptitude tests. They assess your ability to identify logical patterns quickly. Here are some tips for solving number series problems:

• Look at the differences between consecutive terms. If the differences are constant, it's an arithmetic series.
• If the differences are not constant, look at the differences of the differences (second-level differences). As in this problem, they might reveal a pattern.
• Check for ratios between consecutive terms. If the ratio is constant, it's a geometric series.
• Consider squares, cubes, or other powers of numbers, or terms related to Fibonacci or other known sequences.
• Sometimes, the pattern might involve alternating operations (e.g., +2, -1, +2, -1) or multiple interleaved series.
• Practice is key to recognizing different types of number series patterns.