The sequence given is: 81, 27, 9, 3, 1.

This appears to be a geometric sequence where each term after the first is found by multiplying the previous term by a constant ratio.

Let's determine the common ratio: $\text{Common ratio} (r) = \frac{\text{second term}}{\text{first term}} = \frac{27}{81} = \frac{1}{3}$

To verify, let's check if this ratio holds for the entire sequence: $81 \times \frac{1}{3} = 27$ $27 \times \frac{1}{3} = 9$ $9 \times \frac{1}{3} = 3$ $3 \times \frac{1}{3} = 1$

Since the ratio is consistent, we can confirm that this is a geometric sequence with a common ratio of $\frac{1}{3}$.

General Term of the Sequence

The general term $a_n$ of a geometric sequence can be found using: $a_n = a_1 \times r^{(n-1)}$

where:

• $a_1$ is the first term of the sequence (81),
• $r$ is the common ratio ($\frac{1}{3}$),
• $n$ is the term number.

So, the general formula for the nth term of this sequence is: $a_n = 81 \times \left(\frac{1}{3}\right)^{(n-1)}$

Examples:

• For $n = 1$: $a_1 = 81 \times \left(\frac{1}{3}\right)^{(1-1)} = 81 \times 1 = 81$

• For $n = 2$: $a_2 = 81 \times \left(\frac{1}{3}\right)^{(2-1)} = 81 \times \frac{1}{3} = 27$

• For $n = 3$: $a_3 = 81 \times \left(\frac{1}{3}\right)^{(3-1)} = 81 \times \left(\frac{1}{3}\right)^2 = 81 \times \frac{1}{9} = 9$

• For $n = 4$: $a_4 = 81 \times \left(\frac{1}{3}\right)^{(4-1)} = 81 \times \left(\frac{1}{3}\right)^3 = 81 \times \frac{1}{27} = 3$

• For $n = 5$: $a_5 = 81 \times \left(\frac{1}{3}\right)^{(5-1)} = 81 \times \left(\frac{1}{3}\right)^4 = 81 \times \frac{1}{81} = 1$

Do you want more details or have any questions?

Here are some related questions:

1. How do you find the sum of the first $n$ terms in a geometric sequence?
2. What is the sum of the first 10 terms of this sequence?
3. Can you derive the formula for the sum of an infinite geometric series?
4. How would the sequence change if the common ratio was $\frac{1}{2}$ instead of $\frac{1}{3}$?
5. What happens to the sequence if the first term is doubled?
6. How do you determine if a series is convergent or divergent?
7. What are some real-life examples where geometric sequences are applied?
8. Can you explain the difference between arithmetic and geometric sequences?

Tip

When working with geometric sequences, always verify the common ratio by dividing consecutive terms to ensure consistency throughout the sequence.