Arithmetic-Geometric Sequence

Definition

An arithmetic-geometric sequence is a sequence that combines the properties of both an arithmetic sequence and a geometric sequence. It exhibits a linear pattern in the differences between consecutive terms, as well as an exponential pattern in the ratios between consecutive terms.

5 Must Know Facts For Your Next Test

1. The general formula for an arithmetic-geometric sequence is $a_n = a_1 \cdot r^{n-1} + d \cdot \frac{1 - r^{n-1}}{1 - r}$, where $a_1$ is the first term, $r$ is the common ratio, and $d$ is the common difference.
2. Arithmetic-geometric sequences can be used to model real-world situations that exhibit both linear and exponential growth or decay, such as compound interest and population growth.
3. The common ratio $r$ determines whether the sequence is increasing (|$r$| > 1), decreasing (|$r$| < 1), or constant (|$r$| = 1).
4. If $r = 1$, the arithmetic-geometric sequence reduces to a purely arithmetic sequence, and if $d = 0$, it reduces to a purely geometric sequence.
5. Arithmetic-geometric sequences have applications in various fields, including finance, biology, and physics, where they can be used to model phenomena with both linear and exponential components.

Review Questions

• Explain how an arithmetic-geometric sequence differs from a purely arithmetic sequence or a purely geometric sequence.
  • An arithmetic-geometric sequence combines the properties of both an arithmetic sequence and a geometric sequence. Unlike a purely arithmetic sequence, where the difference between consecutive terms is constant, an arithmetic-geometric sequence exhibits a linear pattern in the differences between terms. And unlike a purely geometric sequence, where the ratio between consecutive terms is constant, an arithmetic-geometric sequence has an exponential pattern in the ratios between terms. This combination of linear and exponential components allows arithmetic-geometric sequences to model real-world situations that exhibit both linear and exponential growth or decay.
• Describe the role of the common ratio $r$ and the common difference $d$ in the general formula for an arithmetic-geometric sequence.
  • The common ratio $r$ determines whether the sequence is increasing, decreasing, or constant. If |$r$| > 1, the sequence is increasing; if |$r$| < 1, the sequence is decreasing; and if |$r$| = 1, the sequence is constant. The common difference $d$ represents the linear component of the sequence, as it determines the constant difference between consecutive terms. Together, the common ratio $r$ and the common difference $d$ define the unique characteristics of an arithmetic-geometric sequence and allow it to model a wide range of real-world phenomena.
• Analyze how an arithmetic-geometric sequence can be used to model compound interest, and explain the significance of the common ratio and common difference in this context.
  • Compound interest can be modeled using an arithmetic-geometric sequence, where the common ratio $r$ represents the interest rate, and the common difference $d$ represents the regular deposits or withdrawals. In this context, the common ratio $r$ determines the exponential growth or decay of the account balance, while the common difference $d$ represents the linear component of the sequence, such as periodic contributions or withdrawals. The combination of these two factors allows the arithmetic-geometric sequence to accurately model the complex dynamics of compound interest, which is essential for financial planning and investment strategies.

Related terms

Arithmetic Sequence: A sequence where the difference between any two consecutive terms is constant.

Geometric Sequence: A sequence where the ratio between any two consecutive terms is constant.

Common Difference: The constant difference between consecutive terms in an arithmetic sequence.