Finite Geometric Sequence

Definition

A finite geometric sequence is a sequence where each term is obtained by multiplying the previous term by a common ratio. The sequence has a finite number of terms and is used to model situations with exponential growth or decay.

5 Must Know Facts For Your Next Test

1. The formula for the $n$th term of a finite geometric sequence is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio.
2. The sum of the first $n$ terms of a finite geometric sequence is given by the formula $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$.
3. Finite geometric sequences can be used to model situations with exponential growth, such as compound interest, or exponential decay, such as radioactive decay.
4. The common ratio, $r$, must be greater than 0 and less than 1 for the sequence to represent a decay, and greater than 1 for the sequence to represent growth.
5. Finite geometric sequences have a maximum number of terms, unlike infinite geometric sequences, which continue indefinitely.

Review Questions

• Explain how the formula for the $n$th term of a finite geometric sequence, $a_n = a_1 \cdot r^{n-1}$, is derived.
  • The formula for the $n$th term of a finite geometric sequence, $a_n = a_1 \cdot r^{n-1}$, is derived by recognizing the pattern in the sequence. Each term is obtained by multiplying the previous term by the common ratio, $r$. Therefore, the $n$th term can be expressed as the first term, $a_1$, multiplied by the common ratio raised to the power of $n-1$, since there are $n-1$ multiplications by $r$ to get from the first term to the $n$th term.
• Describe the conditions under which a finite geometric sequence represents exponential growth versus exponential decay.
  • For a finite geometric sequence to represent exponential growth, the common ratio, $r$, must be greater than 1. This means that each term is larger than the previous term, resulting in an overall increasing sequence. Conversely, for a finite geometric sequence to represent exponential decay, the common ratio, $r$, must be between 0 and 1. In this case, each term is smaller than the previous term, resulting in an overall decreasing sequence.
• Explain how the formula for the sum of the first $n$ terms of a finite geometric sequence, $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$, is derived and how it can be used to model real-world situations.
  • The formula for the sum of the first $n$ terms of a finite geometric sequence, $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$, is derived by recognizing the pattern in the sequence and using the formula for the sum of a finite geometric series. This formula can be used to model real-world situations involving exponential growth or decay, such as compound interest, where the sum of the first $n$ terms represents the total amount accumulated over $n$ periods, or radioactive decay, where the sum of the first $n$ terms represents the remaining radioactive material after $n$ time periods.

Related terms

Geometric Sequence: A sequence where each term is obtained by multiplying the previous term by a common ratio.

Common Ratio: The constant ratio between consecutive terms in a geometric sequence.

Infinite Geometric Sequence: A geometric sequence that continues indefinitely, without a finite number of terms.