What is the Next Term in the Geometric Sequence 7, 14, 28?

Geometric sequences are a fundamental concept in mathematics, often appearing in various applications from finance to science. In a geometric sequence, each term after the first is obtained by multiplying the previous term by a fixed, non-zero number known as the common ratio. This article will walk you through finding the next term in the given geometric sequence: 7, 14, 28. We will also discuss the importance of understanding geometric sequences and common ratios in both academic and real-world contexts.

Understanding the Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Let's take a look at the sequence 7, 14, 28.

Identifying the Common Ratio

To determine the common ratio in the sequence 7, 14, 28, we need to divide any term by the previous term. This process will help us find the constant multiplier that defines our sequence. From the first term to the second term:

r frac{14}{7} 2

From the second term to the third term:

r frac{28}{14} 2

We now know that the common ratio is 2. This means that each term in the sequence is obtained by multiplying the previous term by 2.

Finding the Next Term

Once we have identified the common ratio, we can find the next term in the sequence by multiplying the previous term by this common ratio. For the given sequence 7, 14, 28, we need to find the next term after 28.

text{Next term} 28 times 2 56

So, the next term in the sequence is 56.

Additional Examples and Patterns

Understanding geometric sequences goes beyond a single example. Let's look at some additional terms in the sequence 7, 14, 28 to see the pattern more clearly. Term 4: 56 Term 5: 56 Term 6: 56 Term 7: 56 And so on… Each term remains 56 after the third term, as stated in the given examples. As we can see, the sequence stabilizes at 56 after the third term due to the constant multiplication by the common ratio. This stabilization can be a key point in understanding and predicting future terms in the sequence.

Applications of Geometric Sequences

Geometric sequences have numerous real-world applications. They can be used in financial models to describe compound interest, in science to model population growth, and in technology to understand signal transmission and decay. By understanding the common ratio, mathematicians and scientists can make accurate predictions and models based on these sequences. For instance, in finance, geometric sequences can help in calculating compound interest over time. If $7 is invested at an interest rate that doubles the investment every year, the sequence 7, 14, 28, 56, 112, ... represents the value of the investment over consecutive years.

Conclusion

In summary, a geometric sequence is a sequence where each term is found by multiplying the previous term by a fixed, non-zero ratio. In the sequence 7, 14, 28, the common ratio is 2. By multiplying the last known term by the common ratio, we can determine the next term in the sequence, which in this case is 56. Understanding these concepts and patterns can be crucial in various academic and real-world applications. Whether it's in financial models, scientific predictions, or technological advancements, geometric sequences play a vital role in comprehending and solving real problems.

Related Keywords

- Geometric sequence - Common ratio - Next term