Problem 6 - Algebra 2

Question

Is the sequence geometric? If so, find the common ratio and the next two terms. $$ 7,0.7,0.07,0.007, \ldots $$

Step-by-Step Solution

Answer

Yes, the sequence is geometric with a common ratio of 0.1. The next two terms in this sequence are 0.0007 and 0.00007.

Step 1: Identify if the sequence is geometric

A sequence is geometric if each consecutive term can be obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. Therefore, take the ratio between any two consecutive terms and verify if it is the same for all. The ratio between the first two terms (0.7/7), second two terms (0.07/0.7) and third and fourth terms (0.007/0.07), which simplifies to 0.1 in each case. Hence, since the ratio is constant, the sequence is geometric.

Step 2: Determine the common ratio

Having established that this is a geometric sequence, we can confirm that the common ratio is 0.1. This is because each term is obtained by multiplying the previous term by this number.

Step 3: Calculate the next two terms in the sequence

The next term in a geometric sequence is determined by multiplying the last given term by the common ratio. The last given term here is 0.007. Multiplying this term by the common ratio 0.1, we get the next term as 0.0007. This process is repeated in order to get the next term of 0.00007.

Key Concepts

Understanding the Common Ratio in Geometric Sequences

In a geometric sequence, the common ratio is a key element that defines the entire sequence. Simply put, the common ratio is the constant factor by which we multiply each term to get the next one.
For example, in the sequence 7, 0.7, 0.07, 0.007, and so on, the common ratio is 0.1. This means you multiply any term by 0.1 to get the following term.

• The first term (7) multiplied by 0.1 becomes 0.7.
• Then, 0.7 multiplied by 0.1 gives 0.07, and so forth.

Knowing the common ratio allows you to extend the pattern endlessly, making predictions about the future terms easy and systematic. It's important to ensure that this ratio remains the same between all sets of consecutive terms, confirming the sequence's geometric nature.

Recognizing Sequence Patterns

Recognizing sequence patterns is crucial to identifying the type of sequence you're dealing with. In a geometric sequence, you will notice that each term results from multiplying the previous one by a constant, known as the common ratio.
In our example sequence 7, 0.7, 0.07, 0.007, observe how the terms are decreasing consistently.

• This consistent multiplication produces a distinct numeric pattern where each term is a tenth of the previous term.

By spotting this pattern, we can quickly determine the sequence rules. For geometric sequences, this pattern recognition helps confirm the constant relationship between numbers. Understanding and identifying these patterns make it easier to work with sequences in mathematics, giving you strategies to predict upcoming terms and solve related problems.

Exploring Mathematical Sequences

Mathematical sequences are fascinating structures where numbers follow specific rules. There are various types, including arithmetic, geometric, and others.
A sequence is simply a list of numbers in a specific order.
In our context, a geometric sequence is defined by its consistent multiplication pattern among terms.

• Arithmetic sequences, on the other hand, differ as they add a constant value to each term.
• Geometric sequences, like 7, 0.7, 0.07, 0.007, involve multiplying by a common ratio such as 0.1.

This categorization helps mathematicians and students understand how each sequence behaves and its properties. Recognizing the differences between sequence types aids in selecting the right formulas and methods to find a specific term or solve any sequence-related queries.