Evaluate the geometric sequence.           1.Find x so that x, x + 4 and x + 7        

To find the value of \( x \) such that \( x \), \( x + 4 \), and \( x + 7 \) form a geometric sequence, we use the property of geometric sequences: the ratio between consecutive terms must be the same. That is:

\[
\frac{x + 4}{x} = \frac{x + 7}{x + 4}
\]

Now, let's solve this step by step.

1. Cross-multiply to eliminate the denominators:
\[
(x + 4)^2 = x(x + 7)
\]

2. Expand both sides:
\[
(x + 4)(x + 4) = x(x + 7)
\]
\[
x^2 + 8x + 16 = x^2 + 7x
\]

3. Simplify the equation:
\[
x^2 + 8x + 16 = x^2 + 7x
\]
Subtract \( x^2 \) from both sides:
\[
8x + 16 = 7x
\]

4. Solve for \( x \):
\[
8x - 7x = -16
\]
\[
x = -16
\]

Thus, \( x = -16 \).

### Verification:
The terms would be:
- First term: \( x = -16 \)
- Second term: \( x + 4 = -12 \)
- Third term: \( x + 7 = -9 \)

Check the ratio:
\[
\frac{-12}{-16} = \frac{3}{4}, \quad \frac{-9}{-12} = \frac{3}{4}
\]
Since the ratios are equal, the terms form a geometric sequence.

The value of \( x \) is \( -16 \).

---

Would you like more details on any part of the solution?

Here are 5 related questions for practice:

1. Find the value of \( x \) such that \( 2x, 3x, 6x \) form a geometric sequence.
2. Determine \( x \) for the sequence \( x, x+3, x+6 \) to be geometric.
3. Solve for \( x \) if \( x, x-2, x-4 \) is a geometric sequence.
4. Find the common ratio of the geometric sequence \( 5, 15, 45 \).
5. If \( x^2, x^2 + 5, x^2 + 10 \) form a geometric sequence, find \( x \).

**Tip:** In a geometric sequence, the common ratio between terms can be very useful in solving for unknowns. Always check that the ratio remains constant across consecutive terms!

Evaluate the geometric sequence. 1. Find x so that x, x + 4 and x + 7 form a geometric sequence.

Learn how to solve for x in the geometric sequence where the terms are x, x + 4, and x + 7. This problem explores the algebraic concepts of ratios in geometric sequences, cross-multiplication, and equation solving. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation