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 $-16$.

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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!