Understanding the Sequence Pattern

The given sequence is 20, 10, 10, 15, 30, 75, X. To find the value of X, we need to identify the mathematical relationship or pattern between consecutive terms in the sequence.

Identifying the Relationship Between Terms

Let's examine how each term relates to the previous term. We can look at the ratio of a term to its preceding term:

• From 20 to 10: $latex \frac{10}{20} = 0.5$
• From 10 to 10: $latex \frac{10}{10} = 1$
• From 10 to 15: $latex \frac{15}{10} = 1.5$
• From 15 to 30: $latex \frac{30}{15} = 2$
• From 30 to 75: $latex \frac{75}{30} = 2.5$

This shows that each term is obtained by multiplying the previous term by a certain factor.

Discovering the Pattern in Multipliers

The factors by which we multiply each term to get the next one are 0.5, 1, 1.5, 2, and 2.5. Let's look at the difference between these multipliers:

• $latex 1 - 0.5 = 0.5$
• $latex 1.5 - 1 = 0.5$
• $latex 2 - 1.5 = 0.5$
• $latex 2.5 - 2 = 0.5$

The difference between consecutive multipliers is constant, which is 0.5. This indicates that the multipliers themselves form an arithmetic progression with a common difference of 0.5.

Calculating the Value of X

Following this pattern, the next multiplier in the sequence should be $latex 2.5 + 0.5 = 3$.

To find the value of X, we multiply the last term in the sequence (75) by the next multiplier (3):

$latex X = 75 \times 3 = 225$

Summary of the Sequence and Pattern

Here's a table summarizing the sequence, the operation, and the multiplier:

Thus, the value of X that continues the pattern in the sequence is 225.

Revision Table: Key Sequence Pattern Concepts

Understanding sequence patterns often involves looking for relationships like addition, subtraction, multiplication, division, or a combination, sometimes involving the term number or previous terms.

Additional Information: Types of Number Sequences

Number sequences can follow many different rules. Here are a few common types:

• Arithmetic Sequences: Each term after the first is found by adding a constant difference (common difference) to the previous term. Example: 3, 6, 9, 12... (common difference is 3).
• Geometric Sequences: Each term after the first is found by multiplying the previous term by a constant ratio (common ratio). Example: 2, 6, 18, 54... (common ratio is 3).
• Fibonacci Sequence: Each term is the sum of the two preceding terms. Example: 0, 1, 1, 2, 3, 5, 8...
• Other Sequences: Sequences can be based on squares (1, 4, 9, 16...), cubes (1, 8, 27, 64...), prime numbers (2, 3, 5, 7, 11...), or more complex rules like the one in this question, where the multiplier changes in a predictable way.

Analyzing the differences or ratios between terms is a common strategy for solving sequence problems.