Solving the Number Pattern: Find the Missing Term

Let's analyze the given pattern of numbers to find the logic and determine the missing term represented by the question mark (?). The pattern is:

55 63 ? 48 58 40 49 25 49

Observing the sequence, we can see there are 9 numbers. Let's try grouping these numbers into sets of three:

• Group 1: 55, 63, ?
• Group 2: 48, 58, 40
• Group 3: 49, 25, 49

Let's examine the relationship between the numbers within each group, especially focusing on digit-based operations, as these are common in such patterns.

Analyzing Group 2 (48, 58, 40)

Let's look for a relationship between the numbers in Group 2 using their digits.

• First number: 48 (Digits 4, 8)
• Second number: 58 (Digits 5, 8)
• Third number: 40 (Digits 4, 0)

Consider the second number, 58. The product of its digits is \(5 \times 8 = 40\). This is equal to the third number in this group.

Observation 1 (Group 2): The product of the digits of the second number equals the third number.

\(5 \times 8 = 40\)

Analyzing Group 3 (49, 25, 49)

Let's apply a similar digit-based analysis to Group 3.

• First number: 49 (Digits 4, 9)
• Second number: 25 (Digits 2, 5)
• Third number: 49 (Digits 4, 9)

Consider the first number, 49. The difference between its digits is \(|4 - 9| = |-5| = 5\). Squaring this difference gives \(5^2 = 25\). This is equal to the second number in this group.

Observation 2 (Group 3): The square of the difference of the digits of the first number equals the second number.

\((|4 - 9|)^2 = 5^2 = 25\)

Applying Pattern Logic to Group 1 (55, 63, ?)

We have found distinct digit-based rules within Group 2 and Group 3. Let's look for a consistent rule across the groups, or a pattern in the rules themselves.

Let's re-examine the numbers in Group 2 (48, 58, 40). Consider the sum of the digits of the first and third numbers: \((4+8) + (4+0) = 12 + 4 = 16\). The second number is 58. The difference is \(58 - 16 = 42\).

Let's test if this relationship holds for Group 1 (55, 63, ?).

• First number: 55 (Sum of digits = \(5 + 5 = 10\))
• Second number: 63
• Third number: ? (Let the missing number be \(x\))

Assume the pattern for Group 1 is: (Sum of digits of 1st number) + (Sum of digits of 3rd number) + Constant = 2nd number. Using the constant 42 found from Group 2:

\((\text{Sum of digits of } 55) + (\text{Sum of digits of } x) + 42 = 63\)

\(10 + (\text{Sum of digits of } x) + 42 = 63\)

\(52 + (\text{Sum of digits of } x) = 63\)

\(\text{Sum of digits of } x = 63 - 52\)

\(\text{Sum of digits of } x = 11\)

This suggests the missing number \(x\) is a number from the options whose digits sum up to 11.

Checking the Options

Let's calculate the sum of digits for each option:

• Option 1: 49
  Sum of digits = \(4 + 9 = 13\). Does not match 11.
• Option 2: 58
  Sum of digits = \(5 + 8 = 13\). Does not match 11.
• Option 3: 47
  Sum of digits = \(4 + 7 = 11\). Matches 11.
• Option 4: 51
  Sum of digits = \(5 + 1 = 6\). Does not match 11.

The only number from the options with a sum of digits equal to 11 is 47.

Conclusion

Based on the pattern where the sum of digits of the first and third numbers, plus 42, equals the second number (observed in Group 2 and consistently applied to Group 1), the missing number must have a sum of digits equal to 11. Among the given options, only 47 satisfies this condition. Therefore, 47 is the number that replaces the question mark.

The complete pattern with the missing number is: 55 63 47 48 58 40 49 25 49.

Revision Table: Key Pattern Solving Concepts

Understanding different types of number patterns is crucial for quantitative aptitude and reasoning questions. Here are some common types:

• Arithmetic Series: Constant difference between consecutive terms.
• Geometric Series: Constant ratio between consecutive terms.
• Fibonacci Series: Each term is the sum of the two preceding ones.
• Square/Cube Series: Terms are squares or cubes of natural numbers.
• Prime Number Series: Sequence of prime numbers.
• Digit-based Patterns: Patterns involving sum, difference, product, or other operations on the digits of the numbers. This is the type of pattern seen in the problem above.
• Alternating Patterns: Multiple sequences interleaved or rules that alternate.

Additional Information: Approaches to Solving Number Patterns

When faced with a number pattern question, consider these steps:

1. Observe the Sequence: Look at how the numbers are changing - increasing, decreasing, alternating.
2. Find Differences/Ratios: Calculate the difference or ratio between consecutive terms. Look for patterns in these differences or ratios.
3. Look at Positions: Consider the position of each number in the sequence (1st, 2nd, 3rd, ...). Is there a pattern related to the position number?
4. Group Numbers: Sometimes, the pattern is found by grouping numbers (e.g., in pairs, threes, or fours).
5. Check Digit Properties: If simple arithmetic or positional patterns aren't obvious, examine the digits of the numbers. Look at sums, differences, products, or even powers of digits.
6. Look for Alternating Rules: The pattern might involve different rules applied to alternate terms or groups of terms.
7. Test Hypotheses: Once you think you've found a rule, test it on the known parts of the sequence.
8. Use Options: If it's a multiple-choice question, check if the options fit your hypothesized pattern. This can sometimes help identify the rule or confirm your answer.

Complex patterns might combine multiple types of rules, requiring careful observation and systematic testing of possibilities.