Analyzing the Number Series Pattern

This question asks us to identify the pattern in a given series of numbers and find the missing term. The series is 3, 7, 16, 35, ?.

Let's examine the relationship between consecutive terms in the series to uncover the underlying pattern.

Identifying the Series Pattern

We look at the differences or relationships between each pair of numbers:

• From 3 to 7: The difference is \(7 - 3 = 4\). Alternatively, we can see if multiplication is involved. \(3 \times 2 = 6\). Adding 1 gives 7. So, \(3 \times 2 + 1 = 7\).
• From 7 to 16: The difference is \(16 - 7 = 9\). Let's check the multiplication pattern: \(7 \times 2 = 14\). Adding 2 gives 16. So, \(7 \times 2 + 2 = 16\).
• From 16 to 35: The difference is \(35 - 16 = 19\). Checking the multiplication pattern: \(16 \times 2 = 32\). Adding 3 gives 35. So, \(16 \times 2 + 3 = 35\).

We can observe a consistent pattern emerging:

Each term is obtained by multiplying the previous term by 2 and then adding a number that increases sequentially starting from 1.

The pattern for the addition part is +1, +2, +3, ...

Calculating the Missing Term in the Series

Based on the identified pattern, the next number to be added in the sequence is 4.

The last given term is 35.

Applying the pattern:

• Multiply the last term by 2: \(35 \times 2 = 70\).
• Add the next number in the sequence (which is 4): \(70 + 4 = 74\).

So, the missing term in the series is 74.

Verifying the Series Pattern

Let's write out the series generation based on our pattern:

• Starting term: 3
• Second term: \(3 \times 2 + 1 = 7\) (Matches the given series)
• Third term: \(7 \times 2 + 2 = 16\) (Matches the given series)
• Fourth term: \(16 \times 2 + 3 = 35\) (Matches the given series)
• Fifth term (Missing term): \(35 \times 2 + 4 = 74\)

The pattern holds true, and the calculated missing term is 74.

Conclusion for the Number Series Question

The series follows the rule where each term is twice the previous term plus a sequentially increasing number (starting from 1). Applying this pattern to the last given term, 35, results in 74 as the next term.

Revision Table: Understanding Number Series

Understanding different types of number series is crucial for solving such problems. Common types include:

Additional Information: Strategies for Solving Number Series

When tackling number series questions in logic and reasoning sections, consider these strategies:

• Calculate Differences: Find the difference between consecutive terms. Look for a pattern in these differences.
• Calculate Ratios: Find the ratio between consecutive terms, especially if the series grows or shrinks rapidly.
• Look for Multiplication/Division: Check if terms are multiplied or divided by a constant or a varying number.
• Consider Squares and Cubes: Terms might be related to squares (\(n^2\)) or cubes (\(n^3\)) or near squares/cubes (\(n^2 \pm k\)).
• Check for Alternating Patterns: Sometimes, the pattern applies to alternate terms (e.g., 1st, 3rd, 5th terms vs. 2nd, 4th, 6th terms).
• Combine Operations: The pattern might involve a combination of operations (like multiplication and addition, as seen in this question).
• Look for Famous Series: Recognize common series like arithmetic, geometric, Fibonacci, prime numbers, etc.

Practicing different types of series helps in quickly identifying the pattern during exams.