Question

Find x so that x, x + 2, and x + 3 are consecutive terms of a geometric sequence.

Short answer

The value of x is - 4.

Step-by-step solution

Step 1. Given information

Given terms are x, x + 2, and x + 3 are consecutive terms of a

geometric sequence.

Step 2. Equation for geometric sequence.  

The common ratio of geometric sequence is $\frac{a_1}{a_2}=\frac{a_2}{a_3}$

Where $a_1$, $a_2$and $a_3$ are the consecutive terms of Geometric sequence. So, The terms are in common ratio.

$\frac{x}{x+2}=\frac{x+2}{x+3}$

${(x+2)}^2=x(x+3)$

Step 3. Solving the Geometric Proportional ratio equation .

Solving the equation :

$$\begin{gathered} {(x+2)}^2=x(x+3) \\ {x}^2+4x+4=x^2+3x \\ {x}^2+4x+4-4=x^2+3x-4 \\ {x}^2+4x=x^2+3x-4 \\ {x}^2+4x-\left(x^2+3x\right)=x^2+3x-4-\left(x^2+3x\right) \\ x=-4 \end{gathered}$$

Therefore, the value of x is -4 .