Finding the Missing Number in the Series

The question asks us to identify the missing number in the given series: 2, 3, 10, 39, ?, 1175.

To solve this, we need to analyze the relationship between consecutive terms in the number series to find a pattern.

Analyzing the Number Series Pattern

Let's look at the relationship between each term and its preceding term:

• From 2 to 3: \(3 = 2 \times 2 - 1\)
• From 3 to 10: \(10 = 3 \times 3 + 1\)
• From 10 to 39: \(39 = 10 \times 4 - 1\)

Observing these calculations, we can see a pattern emerging. The pattern involves multiplying the previous term by a number that increases sequentially (2, 3, 4, ...) and then either adding 1 or subtracting 1, with the operation alternating (-1, +1, -1).

Identifying the Rule for the Series

The pattern seems to be: Term\(_n\) = Term\(_{n-1}\) \(\times\) n + \((\text{-}1)^{n-1}\)

Let's verify this rule with the given terms:

• For the 2nd term (n=2): Term\(_2\) = Term\(_1\) \(\times\) 2 + \((\text{-}1)^{2-1}\) = \(2 \times 2 + (\text{-}1)^1\) = \(4 - 1 = 3\). This matches the series.
• For the 3rd term (n=3): Term\(_3\) = Term\(_2\) \(\times\) 3 + \((\text{-}1)^{3-1}\) = \(3 \times 3 + (\text{-1})^2\) = \(9 + 1 = 10\). This matches the series.
• For the 4th term (n=4): Term\(_4\) = Term\(_3\) \(\times\) 4 + \((\text{-1})^{4-1}\) = \(10 \times 4 + (\text{-1})^3\) = \(40 - 1 = 39\). This matches the series.

Calculating the Missing Number

The missing number is the 5th term in the series (n=5). Using the pattern:

Term\(_5\) = Term\(_4\) \(\times\) 5 + \((\text{-1})^{5-1}\)

Term\(_5\) = \(39 \times 5 + (\text{-1})^4\)

Term\(_5\) = \(195 + 1\)

Term\(_5\) = \(196\)

Verifying the Next Term

Let's verify if the 6th term (1175) follows the pattern using our calculated 5th term (196):

Term\(_6\) = Term\(_5\) \(\times\) 6 + \((\text{-1})^{6-1}\)

Term\(_6\) = \(196 \times 6 + (\text{-1})^5\)

Term\(_6\) = \(1176 - 1\)

Term\(_6\) = \(1175\)

This matches the last term in the given series, confirming our pattern and the calculated missing number are correct.

Conclusion: The Missing Number

Based on the identified pattern, the missing number in the series 2, 3, 10, 39, ?, 1175 is 196.

The correct option is 196.

Revision Table: Number Series Analysis

Additional Information: Solving Number Series Problems

Number series questions are common in logical reasoning and quantitative aptitude tests. To solve them, you should look for patterns based on:

• Differences between consecutive terms (arithmetic progression, increasing/decreasing differences)
• Ratios between consecutive terms (geometric progression)
• Combinations of arithmetic and geometric operations (like the pattern in this problem)
• Squares or cubes of numbers, possibly with addition or subtraction
• Fibonacci sequence or similar recursive patterns
• Alternating series (different patterns for odd and even positioned terms)

Systematically checking for these common patterns will help you solve most number series questions.