Constant Sequence

Definition

A constant sequence is a sequence in which every term is the same value. In other words, a constant sequence is a sequence where the difference between any two consecutive terms is zero, and the ratio between any two consecutive terms is one.

5 Must Know Facts For Your Next Test

1. A constant sequence can be represented by the formula $a_n = a$, where $a$ is the constant value and $n$ is the term number.
2. The sum of the first $n$ terms of a constant sequence is simply $n$ times the constant value, or $S_n = na$.
3. Constant sequences are a special case of both arithmetic and geometric sequences, where the common difference is zero and the common ratio is one.
4. Constant sequences are often used to model situations where a quantity remains unchanged over time, such as the value of a fixed deposit or the height of a stationary object.
5. The graph of a constant sequence is a horizontal line, as the value of each term is the same.

Review Questions

• Explain how a constant sequence is defined and how it differs from other types of sequences.
  • A constant sequence is a sequence where every term is the same value. This means the difference between any two consecutive terms is zero, and the ratio between any two consecutive terms is one. This distinguishes constant sequences from arithmetic sequences, where the difference between consecutive terms is constant, and geometric sequences, where the ratio between consecutive terms is constant. Constant sequences are a special case of both arithmetic and geometric sequences, where the common difference is zero and the common ratio is one.
• Describe the properties of a constant sequence and how they can be used to calculate the sum of the first $n$ terms.
  • The key properties of a constant sequence are that the difference between any two consecutive terms is zero, and the ratio between any two consecutive terms is one. This means the formula for a constant sequence is $a_n = a$, where $a$ is the constant value and $n$ is the term number. The sum of the first $n$ terms of a constant sequence can be calculated using the formula $S_n = na$, where $S_n$ is the sum and $n$ is the number of terms. This is because each term in the sequence is the same value, so the sum is simply $n$ times that constant value.
• Explain how constant sequences can be used to model real-world situations and discuss the significance of their properties in those applications.
  • Constant sequences are often used to model situations where a quantity remains unchanged over time, such as the value of a fixed deposit or the height of a stationary object. The properties of constant sequences, such as the difference between consecutive terms being zero and the ratio between consecutive terms being one, make them well-suited for these applications. For example, the sum formula $S_n = na$ can be used to calculate the total value of a fixed deposit after $n$ periods, since the deposit amount does not change. Similarly, the horizontal line graph of a constant sequence can be used to visualize the unchanging height of a stationary object. Understanding the unique characteristics of constant sequences is crucial for properly modeling and analyzing these types of real-world scenarios.

Related terms

Sequence:

A sequence is an ordered list of numbers, where each number in the list is called a term.

Arithmetic Sequence:

An arithmetic sequence is a sequence where the difference between any two consecutive terms is constant.

Geometric Sequence:

A geometric sequence is a sequence where the ratio between any two consecutive terms is constant.