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When working with sequences, it's important to identify the type of sequence before proceeding with any calculations. There are two primary types of sequences: arithmetic and geometric. In an arithmetic sequence, there’s a constant difference between consecutive terms, whereas in a geometric sequence, there’s a constant ratio between consecutive terms.

To identify the sequence given in the problem, we look at the first few terms: 5, 15, 45, 135. We need to ascertain if there’s a consistent pattern. By dividing each term by the previous one:

• 15 divided by 5 is 3
• 45 divided by 15 is 3
• 135 divided by 45 is 3

This consistent division result of 3 indicates a geometric sequence. Understanding and identifying the type of sequence is crucial for determining how to predict future terms. This foundation sets the stage for deeper exploration into how the sequence behaves.

In a geometric sequence, the common ratio is the factor by which each term is multiplied to obtain the next term. It remains constant throughout the sequence.

Using the given sequence of 5, 15, 45, 135, we have already identified that this pattern is geometric due to the consistent division result. Here, each term can be obtained by multiplying the previous term by 3. This value, 3, is termed the common ratio.

• Common ratio = 15/5 = 3
• Also, 45/15 = 3
• 135/45 = 3

The common ratio tells us a lot about the sequence, including how quickly it grows if the ratio is greater than 1 or how it diminishes if the ratio is between 0 and 1. Knowing this ratio is pivotal for any further calculations involving the sequence.

Once the type of sequence and the common ratio are identified, calculating subsequent terms becomes straightforward. For a geometric sequence like ours, each term is simply the result of multiplying the previous term by the common ratio.

To find the next two terms of the sequence 5, 15, 45, 135, we follow a systematic approach:

• The fifth term: Multiply the fourth term (135) by the common ratio (3) to get 405.
• The sixth term: Multiply the fifth term (405) by the common ratio (3) to get 1215.

Thus, the sequence continues as 405, 1215, and so on. Understanding this method allows you not only to predict future values but also to navigate and solve real-world problems modeled by geometric sequences.