Analyzing the Number Series Question

The question asks us to find the next number in the given series: 35, 36, 39, 48, ?. To solve this, we need to identify the pattern or rule governing the progression of the numbers in the series.

Identifying the Pattern in the Number Series

Let's look at the differences between consecutive terms in the series:

• Difference between the 2nd and 1st term: \(36 - 35 = 1\)
• Difference between the 3rd and 2nd term: \(39 - 36 = 3\)
• Difference between the 4th and 3rd term: \(48 - 39 = 9\)

The differences we found are 1, 3, and 9. Let's examine these differences for a pattern.

• The first difference is 1.
• The second difference is 3, which is \(1 \times 3\).
• The third difference is 9, which is \(3 \times 3\) or \(1 \times 3 \times 3\).

It appears that each successive difference is obtained by multiplying the previous difference by 3. Alternatively, the differences can be expressed as powers of 3:

• Difference 1: \(1 = 3^0\)
• Difference 2: \(3 = 3^1\)
• Difference 3: \(9 = 3^2\)

This pattern suggests that the differences are successive powers of 3, starting from \(3^0\).

Calculating the Next Term in the Series

Following the identified pattern, the next difference should be the next power of 3, which is \(3^3\).

• Next difference = \(3^3 = 3 \times 3 \times 3 = 27\)

To find the next term in the series (which replaces the question mark), we add this calculated difference to the last term of the series (48).

• Next term = Last term + Next difference
• Next term = \(48 + 27\)
• Next term = \(75\)

Verifying the Solution

The series with the calculated next term is: 35, 36, 39, 48, 75.

Let's check the differences again:

• \(36 - 35 = 1 = 3^0\)
• \(39 - 36 = 3 = 3^1\)
• \(48 - 39 = 9 = 3^2\)
• \(75 - 48 = 27 = 3^3\)

The pattern holds true. The number that replaces the question mark is 75.

Conclusion

Based on the pattern of adding increasing powers of 3 (\(3^0, 3^1, 3^2, 3^3\)...) to the terms, the number that replaces the question mark is 75.

Revision Table: Number Series Patterns

Understanding common number series patterns is key to solving such problems.

• Arithmetic Series: Constant difference between terms (e.g., 2, 4, 6, 8...).
• Geometric Series: Constant ratio between terms (e.g., 2, 4, 8, 16...).
• Difference Series: Pattern in the differences between terms (like this problem). The differences might be an arithmetic series, geometric series, squares, cubes, etc.
• Combination Series: Involves more than one pattern or operation.
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 1, 1, 2, 3, 5, 8...).

Additional Information: Solving Quantitative Reasoning Series

Solving quantitative reasoning series questions often involves a systematic approach:

1. Look at Differences: Calculate the differences between consecutive terms. See if there's a pattern in these differences.
2. Look at Ratios: If the differences aren't consistent, check the ratios between consecutive terms, especially if the numbers are growing or shrinking rapidly.
3. Consider Squares/Cubes: See if the terms are related to squares or cubes of natural numbers, or if the differences are related to them.
4. Check for Alternating Patterns: Sometimes, the pattern alternates between two rules or applies to alternate terms.
5. Combine Operations: The pattern might involve a combination of operations (e.g., multiply by 2 and add 1).
6. Practice: Familiarity with various types of series patterns improves problem-solving speed and accuracy.