Find the 7th term of the geometric sequence shown below.

6, x, squared, comma, minus, 24, x, to the power 6 , comma, 96, x, to the power 10 , comma, point, point, .

6*x*2,−24*x*6,96*x*10,...

To find the 7th term of the geometric sequence, we first need to identify the common ratio, \( r \), of the sequence.

Given sequence: \( 6x^2, -24x^6, 96x^{10}, \dots \)

### Step 1: Determine the common ratio \( r \)

The common ratio \( r \) is found by dividing the second term by the first term:

\[
r = \frac{-24x^6}{6x^2} = \frac{-24}{6} \times \frac{x^6}{x^2} = -4x^4
\]

### Step 2: Verify the common ratio with subsequent terms

Now, let's verify \( r \) by dividing the third term by the second term:

\[
r = \frac{96x^{10}}{-24x^6} = \frac{96}{-24} \times \frac{x^{10}}{x^6} = -4x^4
\]

Since the common ratio is consistent, we can use it to find the 7th term.

### Step 3: Use the formula for the \( n \)-th term of a geometric sequence

The formula for the \( n \)-th term of a geometric sequence is:

\[
a_n = a_1 \times r^{n-1}
\]

For the 7th term \( a_7 \):

\[
a_7 = 6x^2 \times (-4x^4)^{7-1} = 6x^2 \times (-4x^4)^6
\]

### Step 4: Simplify the expression

\[
(-4x^4)^6 = (-4)^6 \times (x^4)^6 = 4096x^{24}
\]

Thus,

\[
a_7 = 6x^2 \times 4096x^{24} = 24576x^{26}
\]

### Final Answer

The 7th term of the geometric sequence is \( \boxed{24576x^{26}} \).

Would you like a more detailed explanation, or do you have any questions? Here are some related questions:

1. How do you find the common ratio in a geometric sequence?
2. What is the formula for the general term of a geometric sequence?
3. How can you verify if a sequence is geometric?
4. What is the 5th term of the same sequence?
5. How does changing the common ratio affect the sequence?

**Tip:** In a geometric sequence, the ratio between consecutive terms remains constant, which is key to solving problems like this one.

Find the 7th term of the geometric sequence shown below: 6x^2, -24x^6, 96x^10, ...

Discover how to find the 7th term of a geometric sequence involving exponents. This detailed guide walks you through the steps of identifying the common ratio and applying the formula for the n-th term, suitable for Grades 9-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation