Understanding the Number Series Pattern

This question asks us to identify the missing number in a given number series: 43, 45, 50, 60, ?, 103.

To find the missing number in a number series, the first step is usually to look for a pattern in the differences between consecutive terms or a pattern in the terms themselves (like squares, cubes, multiples, etc.).

Step-by-Step Analysis of the Series

Let's calculate the differences between the consecutive terms provided:

• Difference between the 2nd and 1st term: \(\text{45 - 43 = 2}\)
• Difference between the 3rd and 2nd term: \(\text{50 - 45 = 5}\)
• Difference between the 4th and 3rd term: \(\text{60 - 50 = 10}\)

The differences we have found are 2, 5, and 10. Let's look for a pattern in these differences.

Identifying the Pattern in Differences

The differences are increasing. Let's see if there's a pattern related to simple arithmetic progression or perhaps squares or cubes.

Consider the possibility that the differences are related to squares of consecutive numbers:

• The first difference, 2, can be expressed as \(\text{1}^2 + 1\).
• The second difference, 5, can be expressed as \(\text{2}^2 + 1\).
• The third difference, 10, can be expressed as \(\text{3}^2 + 1\).

This pattern seems consistent. The difference added to each term is obtained by squaring the position of the difference (starting from 1 for the first difference) and adding 1.

Calculating the Missing Number

Following this pattern, the next difference (to be added to 60) should be based on the 4th position in the sequence of differences. So, the next difference should be \(\text{4}^2 + 1\).

• Next difference = \(\text{4}^2 + 1 = 16 + 1 = 17\)

Now, add this difference to the last known term (60) to find the missing number:

• Missing number = \(\text{60 + 17 = 77}\)

Verifying the Next Term

Let's verify if adding the subsequent difference to 77 gives us the last term in the series, 103. The next difference would be based on the 5th position, so it's \(\text{5}^2 + 1\).

• Next difference after 17 = \(\text{5}^2 + 1 = 25 + 1 = 26\)

Now add this difference to the missing number we found (77):

• \(\text{77 + 26 = 103}\)

This matches the last term in the given series, confirming that our identified pattern and the missing number (77) are correct.

Conclusion

The pattern in the number series is that the difference between consecutive terms is increasing, following the rule \(\text{n}^2 + 1\), where n is the position of the difference (1st difference is \(\text{1}^2+1\), 2nd is \(\text{2}^2+1\), and so on). Using this pattern, the missing number is 77.

The series with the missing number filled in is: 43, 45, 50, 60, 77, 103.

Revision Table: Number Series Patterns

Understanding common patterns is key to solving number series questions quickly. Here are a few types:

• Arithmetic Progression: Constant difference between terms (e.g., 2, 4, 6, 8...)
• Geometric Progression: Constant ratio between terms (e.g., 2, 4, 8, 16...)
• Difference Series: The differences between terms form a simpler series (as seen in this problem).
• Alternating Series: Two different patterns alternate.
• Fibonacci Series: Each term is the sum of the two preceding ones (e.g., 1, 1, 2, 3, 5, 8...).
• Squares/Cubes: Terms are squares or cubes of numbers, possibly with additions or subtractions.

Additional Information: Solving Logical Reasoning Series

When tackling logical reasoning series questions, always start by calculating the differences between consecutive terms. If the differences don't show a simple pattern, calculate the differences between the differences (second-order differences). Look for patterns involving arithmetic progressions, squares, cubes, prime numbers, or alternating operations. Sometimes, the pattern might relate to the position of the term in the series itself. Practice with various types of series helps in quickly recognizing patterns.