Understanding the Number Series Pattern

The question asks us to find the missing number in the given series: 4, 12, 36, 108, 324, ________, 2916.

To solve this, we need to identify the pattern or rule that connects the numbers in the series. Let's look at how each number relates to the one before it.

Identifying the Series Rule

Let's examine the relationship between consecutive terms:

• From 4 to 12: We can see that $4 \times 3 = 12$.
• From 12 to 36: We check if the same multiplication factor applies. $12 \times 3 = 36$. Yes, it does.
• From 36 to 108: Let's check again. $36 \times 3 = 108$. The pattern holds.
• From 108 to 324: Let's confirm. $108 \times 3 = 324$. The pattern is consistent.

It appears that each term in the series is obtained by multiplying the previous term by 3. This type of series is known as a geometric progression where the common ratio is 3.

Calculating the Missing Term in the Series

Following the identified pattern, the missing term is the number after 324. To find it, we must multiply 324 by the common ratio, which is 3.

Missing Term $= 324 \times 3$

Let's perform the multiplication:

So, $324 \times 3 = 972$.

Verifying the Pattern with the Last Term

The series given is 4, 12, 36, 108, 324, ________, 2916.

We found the missing term to be 972. The series would then be: 4, 12, 36, 108, 324, 972, 2916.

Let's check if the pattern holds for the last term. If 972 is the correct missing term, then multiplying it by 3 should give us the next term, which is 2916.

Let's calculate $972 \times 3$:

So, $972 \times 3 = 2916$. This matches the last number provided in the series, confirming that our calculated missing term, 972, is correct.

Conclusion

The pattern in the given number series is that each term is 3 times the previous term. By applying this rule, the missing term is 972.

Revision Table: Number Series Analysis

Additional Information: Types of Number Series

Understanding different types of number series patterns is crucial for solving such problems. Here are a few common types:

• Arithmetic Series: Each term is obtained by adding a constant value (common difference) to the previous term. Example: 2, 5, 8, 11, ... (common difference is 3).
• Geometric Series: Each term is obtained by multiplying the previous term by a constant value (common ratio). The series in this problem is a geometric series. Example: 3, 6, 12, 24, ... (common ratio is 2).
• Square Series: Terms are squares of consecutive numbers. Example: 1, 4, 9, 16, 25, ... ($1^2, 2^2, 3^2, 4^2, 5^2$).
• Cube Series: Terms are cubes of consecutive numbers. Example: 1, 8, 27, 64, ... ($1^3, 2^3, 3^3, 4^3$).
• Fibonacci Series: After the first two terms, each term is the sum of the two preceding ones. Example: 0, 1, 1, 2, 3, 5, 8, ...
• Mixed Series: Patterns might involve a combination of arithmetic and geometric operations, or alternating patterns.

Solving number series problems often involves testing these common patterns first.