Find the missing terms of a geometric sequence of 1, , , ,81

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.

Find the missing terms of a geometric sequence of 1, , , , 81

This page explains how to solve a geometric sequence with missing terms between 1 and 81. Learn how to calculate the common ratio and find the missing terms using key formulas. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation