Understanding the Number Series Problem

The question asks us to identify the number that should replace the question mark (?) in the given sequence of numbers: 5, 13, 29, ?, 125, 253. To solve this, we need to analyze the relationship between consecutive numbers in the series to find a pattern.

Analyzing the Number Series Pattern

Let's look at the differences between the consecutive terms provided in the series:

• Difference between the 2nd and 1st term: $13 - 5 = 8$
• Difference between the 3rd and 2nd term: $29 - 13 = 16$
• Difference between the 6th and 5th term: $253 - 125 = 128$

We can observe a pattern in these differences: 8, 16, 128. It appears that each difference might be doubling to get the next difference.

• $8 \times 2 = 16$
• $16 \times 2 = 32$
• $32 \times 2 = 64$
• $64 \times 2 = 128$

This suggests the sequence of differences might be 8, 16, 32, 64, 128.

Finding the Missing Number

Based on the identified pattern of differences (8, 16, 32, 64, 128), we can find the missing number:

1. The first term is 5.
2. Add the first difference (8) to the first term: $5 + 8 = 13$ (This gives the second term).
3. Add the second difference (16) to the second term: $13 + 16 = 29$ (This gives the third term).
4. Add the third difference (32) to the third term to find the missing term (?): $29 + 32 = 61$.
5. Add the fourth difference (64) to the calculated missing term (61): $61 + 64 = 125$ (This gives the fifth term, which matches the given series).
6. Add the fifth difference (128) to the fifth term (125): $125 + 128 = 253$ (This gives the sixth term, which matches the given series).

The pattern holds true for the entire series when the missing number is 61.

Confirming the Solution

The series with the calculated number is: 5, 13, 29, 61, 125, 253.

The differences are:

• $13 - 5 = 8$
• $29 - 13 = 16$
• $61 - 29 = 32$
• $125 - 61 = 64$
• $253 - 125 = 128$

The differences (8, 16, 32, 64, 128) follow the pattern where each difference is double the previous one. Therefore, the number that replaces the question mark is 61.

Revision Table: Number Series Pattern

Additional Information: Common Number Series Patterns

Number series problems often follow specific patterns. Recognizing these can help solve such questions quickly. Some common patterns include:

• Arithmetic Series: The difference between consecutive terms is constant (e.g., 2, 4, 6, 8...).
• Geometric Series: Each term is found by multiplying the previous term by a constant ratio (e.g., 3, 6, 12, 24...).
• Difference Series: The differences between consecutive terms follow a pattern themselves (as seen in this problem, where differences double). This can extend to differences of differences (second-order differences).
• Square or Cube Series: Terms are related to squares ($1^2, 2^2, 3^2, ...$) or cubes ($1^3, 2^3, 3^3, ...$) or operations involving them (e.g., $n^2 \pm c$).
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5, 8...).
• Mixed Series: The pattern might involve a combination of operations (e.g., alternate addition and subtraction, or a pattern that changes every few terms).

Solving number series questions requires careful observation, calculation, and testing potential patterns.