Math Problem Statement

Find the value of x, if x, x + 4, and .x + 6 are consecutive terms of geometric sequence.

Solution

Given that xx, x+4x + 4, and x+6x + 6 are consecutive terms of a geometric sequence, we know that the ratio between consecutive terms is constant. Therefore, the ratio between the second and first terms should be equal to the ratio between the third and second terms.

Let the common ratio be r.\text{Let the common ratio be } r.

We can set up the equation based on the common ratio:

x+4x=x+6x+4\frac{x + 4}{x} = \frac{x + 6}{x + 4}

To solve for xx, we'll cross-multiply:

(x+4)2=x(x+6)(x + 4)^2 = x(x + 6)

Expanding both sides:

x2+8x+16=x2+6xx^2 + 8x + 16 = x^2 + 6x

Now, subtract x2x^2 from both sides:

8x+16=6x8x + 16 = 6x

Subtract 6x6x from both sides:

2x+16=02x + 16 = 0

Subtract 16 from both sides:

2x=162x = -16

Divide by 2:

x=8x = -8

So, the value of xx is 8\boxed{-8}.

Would you like any further details or have any questions? Here are some related questions:

  1. What is the value of the common ratio rr in this geometric sequence?
  2. How do you verify that x=8x = -8 satisfies the condition of being a geometric sequence?
  3. Can you solve a similar problem with different initial values?
  4. What if the sequence was arithmetic instead of geometric? How would the solution change?
  5. Can you generalize the process for solving problems involving geometric sequences?

Tip: Always verify your solutions by substituting the value back into the original equation to check if the conditions are met.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Geometric Sequences

Formulas

Formula for the common ratio in a geometric sequence

Theorems

-

Suitable Grade Level

Grades 9-12