Solving the Number Series: 64, 67, 76, 103, ?

This question asks us to find the number that replaces the question mark (?) in the given number series: 64, 67, 76, 103, ?.

To solve a number series problem, we typically look for a pattern in the numbers. This pattern could be based on differences, ratios, squares, cubes, or a combination of operations between consecutive terms.

Finding the Pattern in the Number Series

Let's examine the differences between consecutive terms in the series:

• Difference between the 2nd and 1st term: \(67 - 64\)
• Difference between the 3rd and 2nd term: \(76 - 67\)
• Difference between the 4th and 3rd term: \(103 - 76\)

Calculating these differences, we get:

• \(67 - 64 = 3\)
• \(76 - 67 = 9\)
• \(103 - 76 = 27\)

So, the differences between consecutive terms are 3, 9, and 27.

Identifying the Pattern in the Differences

Now let's look at the series of differences: 3, 9, 27. We need to find a pattern in this new series.

We can observe that these numbers are related to the number 3:

• \(3 = 3^1\)
• \(9 = 3 \times 3 = 3^2\)
• \(27 = 3 \times 3 \times 3 = 3^3\)

The pattern in the differences is powers of 3, increasing by one for each step in the original series.

Predicting the Next Number in the Series

Following this pattern, the next difference in the series should be the next power of 3, which is \(3^4\).

Let's calculate \(3^4\):

\(3^4 = 3 \times 3 \times 3 \times 3 = 9 \times 9 = 81\)

The next difference is 81.

To find the next number in the original series, we add this difference (81) to the last term in the series (103).

\(103 + 81\)

Let's calculate the sum:

\(103 + 81 = 184\)

Therefore, the number that replaces the question mark (?) is 184.

Summary of the Series Pattern

The series can be described as starting with 64, and each subsequent term is obtained by adding the next power of 3 to the previous term:

• \(64\)
• \(64 + 3^1 = 64 + 3 = 67\)
• \(67 + 3^2 = 67 + 9 = 76\)
• \(76 + 3^3 = 76 + 27 = 103\)
• \(103 + 3^4 = 103 + 81 = 184\)

Final Answer

The number that replaces the question mark (?) is 184.

Revision Table: Solving Number Series Questions

Additional Information on Number Series Patterns

Number series problems are common in aptitude tests. Recognizing patterns is key. Some common patterns include:

• Arithmetic Series: A constant difference between terms (e.g., 2, 4, 6, 8...).
• Geometric Series: A constant ratio between terms (e.g., 2, 4, 8, 16...).
• Difference Series: The differences between terms follow a pattern, as seen in this problem (e.g., differences are arithmetic, geometric, squares, cubes, etc.).
• Combined Series: More than one pattern or operation is involved (e.g., alternating addition and subtraction, multiplying and adding a constant).
• Fibonacci Series: Each term is the sum of the two preceding ones (e.g., 0, 1, 1, 2, 3, 5...).
• Squares or Cubes: Terms are squares or cubes of numbers, or related to them (e.g., \(1^2, 2^2, 3^2\)... or \(1^3+1, 2^3+1, 3^3+1\)...).

Practicing different types of series helps in quickly identifying the underlying rule during exams.