Have you ever noticed that an object bouncing decreases in height by half with each bounce? This is an example of a geometric sequence, a set of numbers following a specific rule or pattern. A geometric sequence increases or decreases by a constant multiplication or division, also called a geometric progression. Each number in the sequence is known as a term.

Geometric sequences have practical applications in everyday life, like calculating compound interest in finances, population growth, and decay. For instance, the sequence 1, 3, 9, 27, 81... has a common ratio of 3, meaning each number is obtained by multiplying the previous one by 3.

The common ratio, denoted as "r", is the constant factor between each term in a geometric sequence. To find the common ratio, use the term-to-term rule, which involves dividing one term by the one before it.

For example, in the sequence 3, 6, 12, 24, 48..., the common ratio is 2, as each term is obtained by multiplying the previous one by 2. Similarly, the sequence 5, 20, 80, 320, 1280... has a common ratio of 4, and the sequence 32, 16, 8, 4, 2, 1, 0.5... has a common ratio of 0.5.

The term-to-term rule can also find the nth term of a geometric sequence, where "n" is the term's position. To do this, multiply or divide the given term by the common ratio to obtain the next term in the sequence.

For example, in the sequence 8, 40, 200, 1000..., with a common ratio of 5, multiplying 1000 by 5 repeatedly yields the next three terms: 5000, 25000, and 125000. However, it's important to note that as the terms multiply, they rapidly increase or decrease, causing geometric sequences to grow or shrink quickly.

Let's examine an example where the first term is given, and we must find the next terms using the common ratio.

**Example 1:** Find the first five terms of the sequence starting with 13 and with a common ratio of 2.

Multiplying 13 by 2 yields the second term, which is 26. Repeating this process produces the next three terms: 52, 104, and 208. Thus, the first five terms of the sequence are 13, 26, 52, 104, and 208.

**Example 2:** Find the first three terms of the sequence starting with 1000 and a common ratio of 0.25.

To find the next term, divide each term by 4 (the inverse of 0.25). Therefore, the first three terms are 1000, 250, and 62.5.

Arithmetic sequences involve a constant addition or subtraction between terms, whereas geometric sequences involve a constant multiplication or division. For example, in the sequence 2, 4, 6, 8, 10..., the common difference is 2 as each term is obtained by adding 2 to the previous one. This is the primary difference between the two types of sequences.

Now that you understand the fundamentals of geometric sequences, let's test your knowledge with a few practice problems.

- What is the common ratio in the sequence 11, 33, 99, 297...?
- What is the next term in the sequence 9, 18, 36, 72...?
- Find the first four terms of the sequence starting with 25 and with a common ratio of 3.

Solutions:

- The common ratio in the sequence 11, 33, 99, 297... is 3.
- The next term in the sequence 9, 18, 36, 72... is 144.
- The first four terms of the sequence starting with 25 and with a common ratio of 3 are 25, 75, 225, and 675.

**Conclusion:** Geometric sequences follow a specific pattern and have various real-life applications. Understanding the common ratio and the term-to-term rule can help solve problems and identify the next terms in a sequence. Remember, geometric sequences grow or shrink rapidly, so keep this in mind when working with them.

A geometric sequence is a fundamental concept in mathematics that represents repetitive patterns. To understand this concept, it is important to know how to identify and analyze geometric sequences.

To find the next three terms of a geometric sequence, the first step is to determine the common ratio. Once the common ratio is identified, simply multiply it by the previous term to find the next term. For instance, in the sequence 9, 18, 36, 72, 144… the common ratio is 2, and the next three terms are 288, 576, and 1152.

Similarly, if the first term of a geometric sequence is 5 and the common ratio is 4, the first five terms of the sequence would be 5, 20, 80, 320, and 1280.

To determine the term to term rule in a geometric sequence, simply divide one term by the previous term in the sequence. For example, in the sequence 100, 80, 64, 51.2, the term to term rule is 0.8 or 4/5.

- A geometric sequence changes by a constant multiplication or division.
- The constant ratio between terms is known as the common ratio.
- The common ratio can be used to generate terms in the sequence.

A geometric sequence is a type of linear sequence that displays a consistent change in value based on a common ratio. In other words, each term in the sequence is either multiplied or divided by the same number to obtain the next term.

It is essential to identify how the terms are changing in order to distinguish between arithmetic and geometric sequences. If the change between terms is consistent through addition or subtraction, then it is an arithmetic sequence. On the other hand, if the change is consistent through multiplication or division, then it is a geometric sequence.

A finite geometric sequence is one that has a limited number of terms or an ending point. This is distinct from an infinite geometric sequence, which continues indefinitely.

The sum to infinity of a geometric sequence refers to the total of all the terms in the sequence, assuming an infinite number of terms. This calculation plays a crucial role in solving various mathematical problems.

To determine the nth term of a geometric sequence, apply the common ratio repeatedly to the last term until the desired term is reached. This method can be used to predict future terms in the sequence.

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