Finding the Missing Number in the Sequence

The question asks us to find the missing number, X, in the given sequence: 24, X, 12, 18, 36, 90.

To solve this sequence problem, we need to identify the pattern relating consecutive terms. Let's look at the numbers we have after X: 12, 18, 36, 90.

We can examine the relationship between each pair of consecutive terms:

• From 12 to 18: $18 \div 12 = 1.5$. So, $12 \times 1.5 = 18$.
• From 18 to 36: $36 \div 18 = 2$. So, $18 \times 2.0 = 36$.
• From 36 to 90: $90 \div 36 = 2.5$. So, $36 \times 2.5 = 90$.

We can see a clear pattern in the multipliers used to get from one term to the next: 1.5, 2.0, 2.5. These multipliers are increasing by 0.5 each time.

Assuming this pattern continues throughout the sequence, we can work backward to find the value of X.

The multiplier used to get from X to 12 should be the number before 1.5 in the sequence of multipliers. Since the multipliers increase by 0.5 each time, the multiplier before 1.5 would be $1.5 - 0.5 = 1.0$.

So, $X \times 1.0 = 12$. This tells us that $X = 12$.

Now, let's check if this fits the first part of the sequence (24 to X). The multiplier used to get from 24 to X should be the number before 1.0 in the sequence of multipliers. This would be $1.0 - 0.5 = 0.5$.

So, $24 \times 0.5 = X$. This also gives us $X = 12$.

The number 12 fits perfectly into the sequence, maintaining the pattern of multipliers increasing by 0.5.

Let's verify the complete sequence with X = 12:

24, 12, 12, 18, 36, 90

Let's check the multipliers between consecutive terms:

• 24 to 12: $12 \div 24 = 0.5$ ($24 \times 0.5 = 12$)
• 12 to 12: $12 \div 12 = 1.0$ ($12 \times 1.0 = 12$)
• 12 to 18: $18 \div 12 = 1.5$ ($12 \times 1.5 = 18$)
• 18 to 36: $36 \div 18 = 2.0$ ($18 \times 2.0 = 36$)
• 36 to 90: $90 \div 36 = 2.5$ ($36 \times 2.5 = 90$)

The multipliers are 0.5, 1.0, 1.5, 2.0, 2.5, which confirms the pattern of increasing by 0.5 each time. Therefore, the missing number X is 12.

The options provided are:

1. 18
2. 12
3. 9
4. 6

Our calculated value for X is 12, which matches Option 2.

Summary of Sequence Pattern

The pattern in the sequence is that each term is obtained by multiplying the previous term by a multiplier that increases by 0.5 for each step in the sequence.

From the table, we can see that if X = 12, the pattern holds correctly.

Revision Table: Sequence Patterns

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

Additional Information: Analyzing Number Sequences

When faced with a number sequence problem, consider these steps:

1. Look at the Differences: Calculate the difference between consecutive terms. Is it constant (arithmetic sequence)? Do the differences form a new, recognizable sequence?
2. Look at the Ratios: Calculate the ratio between consecutive terms (divide a term by the previous one). Is it constant (geometric sequence)? Do the ratios form a new, recognizable sequence?
3. Look for Combined Operations: Is there a combination of addition/subtraction and multiplication/division? (e.g., multiply by a number and then add/subtract a number).
4. Consider Positions: Does the pattern relate to the position of the term in the sequence (e.g., the nth term is given by a formula involving n)?
5. Look for Squares, Cubes, or Other Powers: Are the terms related to perfect squares, cubes, or sequences based on powers?
6. Check for Alternating Patterns: Sometimes operations alternate (e.g., +2, $\times 3$, +2, $\times 3$).

In this specific problem, the pattern in the ratios between consecutive terms was the key to finding the missing number X in the sequence.