Solving the Number Series: 7, 16, 40, 84, _, 252

This question asks us to find the missing number in the given number series: 7, 16, 40, 84, _____, 252. To solve a number series problem, we need to identify the underlying pattern or rule that governs the sequence of numbers.

Identifying the Pattern in the Number Series

Let's examine the differences between consecutive terms in the series. This is a common strategy for finding the pattern in numerical sequences.

Step 1: Calculate First Differences

Subtract each term from the subsequent term:

• Difference between 16 and 7: ${16 - 7 = 9}$
• Difference between 40 and 16: ${40 - 16 = 24}$
• Difference between 84 and 40: ${84 - 40 = 44}$

The first differences are 9, 24, and 44.

Step 2: Calculate Second Differences

Now, let's look at the differences between these first differences:

• Difference between 24 and 9: ${24 - 9 = 15}$
• Difference between 44 and 24: ${44 - 24 = 20}$

The second differences are 15 and 20.

Step 3: Identify the Pattern in Second Differences

The second differences are 15 and 20. The difference between these is ${20 - 15 = 5}$. This suggests that the second differences form an arithmetic progression with a common difference of 5.

Step 4: Predict the Next Differences

Assuming the pattern continues, the next second difference should be ${20 + 5 = 25}$.

This next second difference (25) is added to the last first difference (44) to find the next first difference: ${44 + 25 = 69}$.

This next first difference (69) is added to the last term in the series (84) to find the missing term: ${84 + 69 = 153}$.

Step 5: Verify the Pattern

Let's see if this pattern holds for the final term, 252. The next second difference would be ${25 + 5 = 30}$. The first difference after 69 would be ${69 + 30 = 99}$. Adding this to our calculated missing term (153): ${153 + 99 = 252}$. This matches the last number in the given series, confirming our pattern is correct.

The Missing Number

Following the pattern of second differences increasing by 5, the missing number in the series is 153.

Conclusion

The series follows a pattern where the differences between consecutive terms increase by an arithmetic progression with a common difference of 5. By applying this pattern, we determined the missing term.

Revision Table: Number Series Pattern Analysis

Additional Information: Types of Number Series Patterns

Number series questions in reasoning sections often follow various patterns. Understanding these can help in solving such problems quickly. Common patterns include:

• Arithmetic Series: Each term is obtained by adding a constant value (common difference) to the previous term.
• Geometric Series: Each term is obtained by multiplying the previous term by a constant value (common ratio).
• Difference Series: The differences between consecutive terms follow a pattern, which could be arithmetic, geometric, or another type of sequence (as seen in this problem with second differences).
• Ratio Series: The ratio between consecutive terms follows a pattern.
• Mixed Series: A combination of two or more patterns, often alternating.
• Fibonacci or Similar Series: Each term is the sum of the two preceding terms (or a variation thereof).
• Square or Cube Series: Terms are squares or cubes of natural numbers, possibly with additions or subtractions.

Analyzing differences (first, second, third, etc.) is a powerful technique to uncover patterns, especially when the relationship isn't immediately obvious.