https://file.aiquickdraw.com/mathbot/file/1726510382643gr0smizs.jpg

To solve the question shown in the image, let's analyze each statement related to the geometric sequence \(6, 18, 54, \ldots\):

**Finding the common ratio:**
From the sequence, the common ratio \(r\) can be determined by dividing any term by the previous one:
\[
r = \frac{18}{6} = 3
\]

**Statement B: "The common ratio is 3."**
- **True**, as calculated above.

**Finding specific terms:**
The \(n\)-th term of a geometric sequence can be calculated using the formula:
\[
a_n = a \cdot r^{(n-1)}
\]
where \(a\) is the first term.

**For the 4th term:**
\[
a_4 = 6 \cdot 3^{(4-1)} = 6 \cdot 27 = 162
\]

**Statement D: "The 4th term is -3 times 54."**
- **False**, as \( -3 \times 54 = -162 \) but \( a_4 = 162 \).

**For the 6th term:**
\[
a_6 = 6 \cdot 3^{(6-1)} = 6 \cdot 243 = 1458
\]

**Statement C: "The 6th term is -1458."**
- **False**, as \( a_6 = 1458 \) and not \(-1458\).

**Comparison of the powers of -3:**
\[
6(-3)^{11} \text{ and } 6(-3)^{10}
\]
Since the base is negative, \( (-3)^{11} \) will be negative, and \( (-3)^{10} \) will be positive. The magnitude will be greater when the power is 11 because the base, 3, is being multiplied one more time.

**Statement A: "6(-3)^{11} is smaller than 6(-3)^{10}"**
- **True**, as \(6(-3)^{11} = 6 \times -177147 \) which is indeed smaller than \( 6 \times 59049 \).

**Correct Statements:**
- **A**: True
- **B**: True

Would you like to delve into any particular detail or do you have any other questions? Here are some related questions:

1. How would you find the 8th term in this sequence?
2. What is the sum of the first 6 terms of this sequence?
3. How does changing the common ratio affect the sequence?
4. Can you explain how negative powers influence the terms in a geometric sequence?
5. What would the sequence look like if the common ratio were fractional, such as 1/3?

**Tip:** In a geometric sequence, each term is derived by multiplying the

Consider the geometric sequence 6, -18, 54, ... Select all that apply:
A) 6(-3)^{11} is smaller than 6(-3)^{10}
B) The common ratio is 3
C) The 6th term is -1458
D) The 4th term is -3 times 54

Explore the geometric sequence starting from 6, -18, 54,... and learn how to determine the common ratio, calculate specific terms using geometric formulas, and analyze the comparison of exponents. Detailed explanation for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation