Let's analyze the given number series to find the pattern and determine the missing number.

The series is: 4, 11, 19, 41, 79, ?, 319

We need to identify the rule that transforms each number into the next number in the sequence.

Discovering the Pattern in the Number Series

Let's look at the differences between consecutive terms:

• \(11 - 4 = 7\)
• \(19 - 11 = 8\)
• \(41 - 19 = 22\)
• \(79 - 41 = 38\)

The differences (7, 8, 22, 38) don't immediately reveal a simple arithmetic progression or a clear pattern.

Let's try looking for a pattern involving multiplication and addition/subtraction.

• From 4 to 11: \(4 \times 2 = 8\). We need to add 3 to get 11. So, \(4 \times 2 + 3 = 11\).
• From 11 to 19: \(11 \times 2 = 22\). We need to subtract 3 to get 19. So, \(11 \times 2 - 3 = 19\).
• From 19 to 41: \(19 \times 2 = 38\). We need to add 3 to get 41. So, \(19 \times 2 + 3 = 41\).
• From 41 to 79: \(41 \times 2 = 82\). We need to subtract 3 to get 79. So, \(41 \times 2 - 3 = 79\).

The pattern appears to be multiplying the current term by 2 and then alternately adding 3 and subtracting 3.

The sequence of operations on the multiplied result is: +3, -3, +3, -3, ...

Applying the Pattern to Find the Missing Number

The last operation applied was subtracting 3 (to get 79). So, for the next term (the missing number), we should multiply the current term (79) by 2 and then add 3.

Missing number \( = 79 \times 2 + 3 \)

Let's calculate this:

\(79 \times 2 = 158\)

\(158 + 3 = 161\)

So, the missing number is 161.

Verifying the Pattern with the Next Term

Let's check if applying the next step in the pattern to 161 gives the subsequent number in the series (319). The next operation after adding 3 should be subtracting 3.

\(161 \times 2 - 3\)

Let's calculate this:

\(161 \times 2 = 322\)

\(322 - 3 = 319\)

This matches the last number in the given series (319). Therefore, the pattern is confirmed, and the missing number is indeed 161.

The series with the missing number filled in is: 4, 11, 19, 41, 79, 161, 319.

Revision Table: Understanding Number Series

Additional Information: Types of Number Series Patterns

Number series questions often involve different types of patterns:

• Arithmetic Progression: A constant difference between consecutive terms (e.g., 2, 4, 6, 8...).
• Geometric Progression: A constant ratio between consecutive terms (e.g., 3, 9, 27, 81...).
• Difference Series: The differences between consecutive terms follow a pattern (e.g., 2, 3, 5, 8, 12... where differences are 1, 2, 3, 4...).
• Alternating Series: Two different patterns alternate within the same series (e.g., 1, 5, 2, 6, 3, 7...).
• Mixed Operations Series: The pattern involves a combination of different operations like the one in this question (multiply and then add/subtract).
• Fibonacci or Similar Series: Each term is the sum of the previous two terms (e.g., 0, 1, 1, 2, 3, 5...).

Solving number series problems involves testing various potential patterns until the correct one is found and confirmed across the given terms.