Finding the Missing Number in the Series

Let's analyze the given number series to find the missing term: 3, 12, __________, 21612.

We need to identify the pattern or rule that governs the sequence. Let's examine the relationship between consecutive terms.

• The first term is 3.
• The second term is 12.

Let's try to find a pattern that generates the terms. We can explore relationships involving multiplication, addition, powers, or a combination of these operations, possibly related to the previous terms.

Consider the possibility that each term is generated based on the preceding terms. Let $T_n$ represent the nth term in the series.

• $T_1 = 3$
• $T_2 = 12$

Let's test a recursive pattern often found in such series. Could the third term ($T_3$) be related to $T_1$ and $T_2$ in a specific way?

Consider the pattern where a term is the square of the previous term plus the term before that. Let's test this hypothesis:

Hypothesized Pattern: $T_n = T_{n-1}^2 + T_{n-2}$ for $n \ge 3$.

Let's apply this pattern to find the third term ($T_3$):

• Here, $n=3$.
• $T_{n-1} = T_{3-1} = T_2 = 12$
• $T_{n-2} = T_{3-2} = T_1 = 3$

Using the hypothesized pattern:

$$T_3 = T_2^2 + T_1$$

$$T_3 = 12^2 + 3$$

$$T_3 = 144 + 3$$

$$T_3 = 147$$

So, the third term according to this pattern is 147. This matches one of the given options.

Let's verify this pattern by calculating the fourth term ($T_4$) using the calculated $T_3 = 147$ and $T_2 = 12$:

$$T_4 = T_3^2 + T_2$$

$$T_4 = 147^2 + 12$$

$$T_4 = (147 \times 147) + 12$$

$$T_4 = 21609 + 12$$

$$T_4 = 21621$$

The calculated fourth term (21621) is very close to the fourth term given in the series (21612), suggesting this is the intended pattern to find the missing number, despite a small difference in the final term listed.

Therefore, the missing number in the series following the pattern $T_n = T_{n-1}^2 + T_{n-2}$ is 147.

Revision Table: Number Series Pattern

Additional Information on Number Series and Patterns

Number series questions are common in logical reasoning and aptitude tests. They assess your ability to identify mathematical or logical patterns in a sequence of numbers. Common types of patterns include:

• Arithmetic Progression: Adding or subtracting a constant value.
• Geometric Progression: Multiplying or dividing by a constant value.
• Differences/Ratios of Differences: Patterns in the differences between consecutive terms.
• Powers/Cubes: Terms related to squares, cubes, or other powers.
• Recursive Relations: Terms depending on one or more preceding terms (like the pattern seen in this question, similar to Fibonacci but with different operations).
• Alternating Patterns: Two different patterns applied alternately.

Solving number series puzzles often involves trying out different potential patterns and performing calculations to see which rule consistently applies to the given terms and predicts the missing term or the subsequent terms.