Discovering the Equation for the Sequence 5, 10, 20

When faced with a sequence like 5, 10, 20, the first step is to identify the pattern that connects the terms. Understanding this pattern allows us to represent the sequence mathematically, which is particularly useful for predicting future terms or solving deeper mathematical problems.

Identifying the Pattern

Let's consider the sequence: 5, 10, 20. In order to identify the pattern, we examine how each term relates to the preceding one:

The first term is 5. The second term is 10, which is 5 multiplied by 2. The third term is 20, which is 10 multiplied by 2, or equivalently, 5 multiplied by 4.

From these observations, it is clear that each term in the sequence is obtained by multiplying the previous term by 2. This can be generalized as each term being a product of the initial term (5) and successive powers of 2.

General Form of the Sequence

Mathematically, we can express the general form of the sequence as:

an 5 x 2n-1

Here, an represents the nth term of the sequence, and n is the position of the term in the sequence (1 for the first term, 2 for the second term, and so on).

Verification

To verify the equation, we can test it with the first few terms of the sequence:

For n 1: a1 5 x 21-1 5 x 20 5, which matches the first term of the sequence. For n 2: a2 5 x 22-1 5 x 21 10, which matches the second term of the sequence. For n 3: a3 5 x 23-1 5 x 22 20, which matches the third term of the sequence.

Since the generated terms match the given sequence, we have verified that the equation is correct.

Alternative Representations

While the previous representation is accurate, there are other ways to represent the same sequence, such as:

5 x 2n or 5 x 2n-1

Both of these denote the same sequence, where n is an integer (0, 1, 2, etc.). The choice between 2n-1 and 2n depends on the starting point of the sequence. If we consider 5 as the first term, then n 0. If we consider 5 as the second term, then n 1.

Conclusion

Understanding the pattern and forming the equation is a fundamental skill in mathematics, providing a structured approach to solve more complex problems. The general form of the sequence, an 5 x 2n-1, is a concise and powerful representation that allows us to predict and calculate any term in the sequence.