Finding the Missing Number in the Series 55, 63, ?, 343, 855, 1855

This question asks us to identify the pattern in the given number series and find the missing number represented by the question mark (?). The given series is:

55, 63, ?, 343, 855, 1855

Let's analyze the differences between consecutive terms in the series to find a potential pattern.

• Difference between the second and first term: $63 - 55 = 8$
• Difference between the fifth and fourth term: $855 - 343 = 512$
• Difference between the sixth and fifth term: $1855 - 855 = 1000$

The observed differences are 8, 512, and 1000. Let's examine these numbers more closely. They appear to be perfect cubes:

• $8 = 2 \times 2 \times 2 = 2^3$
• $512 = 8 \times 8 \times 8 = 8^3$
• $1000 = 10 \times 10 \times 10 = 10^3$

The bases of these cubes are 2, 8, and 10. There seems to be a pattern in these bases: 2, ?, 8, 10. If the bases are increasing even numbers, the sequence of bases could be 2, 4, 6, 8, 10.

Let's assume the pattern of differences is the cube of consecutive even numbers, starting from 2:

• First difference: $2^3 = 8$
• Second difference: $4^3 = 64$
• Third difference: $6^3 = 216$
• Fourth difference: $8^3 = 512$
• Fifth difference: $10^3 = 1000$

Now, let's apply this pattern of differences to the given series:

• Term 1: 55
• Term 2: $55 + 2^3 = 55 + 8 = 63$ (Matches the given second term)
• Term 3: $63 + 4^3 = 63 + 64 = 127$ (This is our potential missing number)
• Term 4: $127 + 6^3 = 127 + 216 = 343$ (Let's check if this matches the given fourth term)

The calculated fourth term is 343, which exactly matches the given fourth term in the series.

Let's continue to verify the pattern with the remaining terms:

• Term 5: $343 + 8^3 = 343 + 512 = 855$ (Matches the given fifth term)
• Term 6: $855 + 10^3 = 855 + 1000 = 1855$ (Matches the given sixth term)

The pattern holds true for the entire series when the missing number is 127. The pattern is that each subsequent term is obtained by adding the cube of the next consecutive even number to the previous term, starting with $2^3$.

Step-by-Step Calculation

1. Identify the given series: 55, 63, ?, 343, 855, 1855.
2. Calculate the differences between known consecutive terms:
   • $63 - 55 = 8$
   • $855 - 343 = 512$
   • $1855 - 855 = 1000$
3. Recognize these differences as cubes of even numbers:
   • $8 = 2^3$
   • $512 = 8^3$
   • $1000 = 10^3$
4. Hypothesize that the differences are cubes of consecutive even numbers (2, 4, 6, 8, 10).
5. Use the hypothesized pattern to find the missing term (Term 3):
   • Term 3 = Term 2 + $4^3$
   • Term 3 = $63 + 64 = 127$
6. Verify the pattern with the subsequent terms using the calculated missing number (127):
   • Term 4 = Term 3 + $6^3 = 127 + 216 = 343$ (Matches the given Term 4)
   • Term 5 = Term 4 + $8^3 = 343 + 512 = 855$ (Matches the given Term 5)
   • Term 6 = Term 5 + $10^3 = 855 + 1000 = 1855$ (Matches the given Term 6)
7. The pattern is confirmed, and the missing number is 127.

Therefore, the number that replaces the question mark is 127.

Revision Table: Series Pattern Analysis

Additional Information: Number Series and Patterns

Number series questions are common in logical reasoning and quantitative aptitude tests. They require you to identify a specific pattern or rule that governs the sequence of numbers. Once the pattern is identified, you can predict the next number in the series or find a missing number.

Common types of patterns include:

• Arithmetic Series: A constant difference is added or subtracted between consecutive terms (e.g., 2, 5, 8, 11,... where the difference is +3).
• Geometric Series: A constant ratio is multiplied or divided between consecutive terms (e.g., 3, 6, 12, 24,... where the ratio is ×2).
• Difference Series: The difference between consecutive terms forms a pattern itself (like in this problem, where the differences are cubes of even numbers). The differences could follow an arithmetic, geometric, or other pattern.
• Mixed Series: A combination of two or more patterns (e.g., alternating arithmetic and geometric progressions, or a pattern involving addition and multiplication).
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5, 8,...).
• Series based on Squares or Cubes: Terms are related to squares or cubes of numbers (e.g., 1, 4, 9, 16,... are squares of 1, 2, 3, 4). The pattern can involve adding/subtracting squares/cubes or the terms themselves being squares/cubes with some modification.

To solve number series problems effectively, it's helpful to:

• Calculate differences between consecutive terms.
• Calculate ratios between consecutive terms.
• Look for squares, cubes, or other powers.
• Check for alternating patterns.
• Consider combinations of basic operations.

Practice with various types of series helps in quickly recognizing patterns during exams.