Common ratio

Definition

The common ratio is a constant factor that is used to multiply each term in a geometric sequence to get the next term. It plays a critical role in understanding how geometric sequences grow or shrink. In essence, if you take any term in the sequence and divide it by the previous term, you will always arrive at the same value, which is the common ratio.

5 Must Know Facts For Your Next Test

1. The common ratio can be positive or negative, influencing whether the terms of the sequence increase or oscillate.
2. To find the common ratio in a geometric sequence, divide any term by its preceding term.
3. The nth term of a geometric sequence can be expressed using the formula: $$a_n = a_1 imes r^{(n-1)}$$ where 'r' is the common ratio.
4. In a series derived from a geometric sequence, such as an infinite geometric series, the sum can be calculated only if the absolute value of the common ratio is less than one.
5. Common ratios that are fractions between 0 and 1 will lead to a sequence that decreases towards zero.

Review Questions

• How does the common ratio determine the nature of a geometric sequence?
  • The common ratio directly influences how the terms of a geometric sequence behave. If the common ratio is greater than one, each term will increase exponentially. Conversely, if the common ratio is between 0 and 1, the terms will decrease and approach zero. A negative common ratio will cause the terms to alternate in sign, creating an oscillating pattern. Thus, understanding the common ratio helps predict whether a sequence will grow, shrink, or oscillate.
• Discuss how you would calculate the sum of an infinite geometric series when given a common ratio.
  • To calculate the sum of an infinite geometric series, it's essential first to confirm that the absolute value of the common ratio is less than one (|r| < 1). The formula for calculating this sum is $$S = \frac{a_1}{1 - r}$$ where 'S' represents the sum, 'a_1' is the first term of the series, and 'r' is the common ratio. This formula shows how even with infinitely many terms, we can find a finite sum based on these two parameters.
• Evaluate how changes in the common ratio affect real-world applications such as population growth or investment returns.
  • In real-world applications like population growth or investment returns, changes in the common ratio can significantly impact long-term outcomes. For example, a higher common ratio reflects rapid growth or high returns on investments leading to exponential increases over time. On the other hand, a lower common ratio may indicate slower growth rates or diminishing returns. By analyzing these ratios, we can make informed predictions about future trends and potential challenges in areas such as finance and ecology.

Related terms

Geometric Sequence: A sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio.

First Term: The initial term of a geometric sequence, often represented as 'a' or 'a_1', which, along with the common ratio, helps define the entire sequence.

Exponential Growth: A pattern of data that shows greater increases over time, often represented in geometric sequences where the common ratio is greater than one.