The given geometric sequence is: $1, \_, \_, \_, 81$.

To find the missing terms in a geometric sequence, we need to determine the common ratio $r$ and use it to fill in the missing terms.

Step 1: General form of a geometric sequence

The terms of a geometric sequence follow the pattern: $a, a \cdot r, a \cdot r^2, a \cdot r^3, \dots$ where $a$ is the first term, and $r$ is the common ratio.

Here, the first term is $a = 1$, and the fifth term is $81$.

Step 2: Relationship between the terms

In a geometric sequence, each term is the previous term multiplied by the common ratio $r$. Therefore: $81 = 1 \cdot r^4$ This simplifies to: $r^4 = 81$ Taking the fourth root of both sides: $r = \sqrt[4]{81} = 3$

Step 3: Finding the missing terms

Now that we know the common ratio $r = 3$, we can calculate the missing terms:

• Second term: $1 \cdot 3 = 3$
• Third term: $1 \cdot 3^2 = 9$
• Fourth term: $1 \cdot 3^3 = 27$

Final Sequence:

$1, 3, 9, 27, 81$

The missing terms are $3, 9, 27$.

Would you like more details or have any questions?

Here are five related questions for further exploration:

1. What is the formula for the $n$-th term of a geometric sequence?
2. How would the sequence change if the first term was a different number?
3. What happens if the common ratio $r$ is a fraction?
4. Can a geometric sequence have negative terms?
5. How do you find the sum of the first $n$ terms of a geometric series?

Tip: In a geometric sequence, the common ratio can be determined from any two consecutive terms by dividing one by the other.