Understanding the Number Series Problem

The question asks us to find the missing number in the given series: 9, 10, 22, 69, 280, ?. This is a common type of question in logical reasoning and quantitative aptitude that requires identifying the underlying pattern connecting consecutive terms in the series.

Discovering the Number Series Pattern

Let's examine the relationship between each pair of consecutive numbers in the series to identify the pattern:

• From the first term (9) to the second term (10): The difference is 10 - 9 = 1. Let's try multiplication and addition/subtraction. If we multiply 9 by 1 and add 1, we get $9 \times 1 + 1 = 9 + 1 = 10$. This fits the second term.
• From the second term (10) to the third term (22): The difference is 22 - 10 = 12. Let's test if the pattern involves multiplying by the next integer (2) and adding the same integer. If we multiply 10 by 2 and add 2, we get $10 \times 2 + 2 = 20 + 2 = 22$. This fits the third term.
• From the third term (22) to the fourth term (69): The difference is 69 - 22 = 47. Let's test if the pattern continues with multiplying by 3 and adding 3. If we multiply 22 by 3 and add 3, we get $22 \times 3 + 3 = 66 + 3 = 69$. This fits the fourth term.
• From the fourth term (69) to the fifth term (280): The difference is 280 - 69 = 211. Let's test if the pattern continues with multiplying by 4 and adding 4. If we multiply 69 by 4 and add 4, we get $69 \times 4 + 4 = 276 + 4 = 280$. This fits the fifth term.

The pattern is consistent: each term is obtained by multiplying the previous term by an integer that increases by 1 starting from 1, and then adding the same integer.

Applying the Pattern to Find the Next Number

Based on the identified pattern, to find the next number after 280, we need to follow the sequence of operations. The last operation involved multiplying by 4 and adding 4. The next operation should involve multiplying by 5 and adding 5.

Let the missing number be ?. We apply the pattern to the last known term (280):

Next term = Previous term $\times$ 5 + 5

? = $280 \times 5 + 5$

First, calculate the multiplication:

$280 \times 5 = 1400$

Next, perform the addition:

$1400 + 5 = 1405$

So, the number that replaces the question mark is 1405.

Conclusion

The series follows the pattern of multiplying the previous term by n and adding n, where n starts at 1 and increments by 1 for each subsequent term. Applying this pattern, the next number in the series 9, 10, 22, 69, 280, ? is 1405.

Revision Table: Number Series Patterns

Understanding common number series patterns is key to solving these types of problems. Here's a quick review of some common patterns:

• Arithmetic Series: A constant difference is added or subtracted between consecutive terms (e.g., 2, 5, 8, 11... difference is +3).
• Geometric Series: Each term is multiplied or divided by a constant ratio to get the next term (e.g., 3, 6, 12, 24... ratio is $\times$ 2).
• Difference Series: The differences between consecutive terms form a pattern (e.g., 1, 2, 4, 7, 11... differences are +1, +2, +3, +4...).
• Product/Ratio Series: Terms are related by multiplication or division, sometimes involving increasing or decreasing factors.
• Mixed Operations Series: Patterns involving a combination of arithmetic and geometric operations, as seen in this problem (Multiply by n, Add n).
• Fibonacci or Similar Series: Each term is the sum of the previous two terms (e.g., 0, 1, 1, 2, 3, 5...). Variations include summing previous three terms, etc.

Additional Information: Solving Logical Series Questions

To effectively solve logical series questions, consider the following tips:

• Look for simple arithmetic differences or ratios first.
• If simple patterns aren't visible, examine the differences between consecutive terms. Sometimes, the differences themselves form a series with a simple pattern.
• Consider patterns involving alternating operations (e.g., +2, -1, +2, -1).
• Look for patterns involving squares, cubes, prime numbers, or Fibonacci sequences.
• Test combinations of operations, like multiplication followed by addition or subtraction.
• Write down the operations/differences between terms clearly to help spot the pattern.
• Practice with various types of series questions to become familiar with common patterns.