Question

Find x so that x - 1, x, and x + 2 are consecutive terms of a geometric sequence.

Short answer

The value of x is 2 .

Step-by-step solution

Step 1. Given information

x - 1, x, and x + 2 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.

Therefore, $$\begin{gathered} \frac{x}{x-1}=\frac{x+2}{x} \\ {x}^2=(x+2)(x-1) \end{gathered}$$

Step 3. Solving the Geometric Proportional ratio equation .

Solving the equation :-

$$\begin{gathered} x^2=x^2+x-2 \\ -x=-2 \end{gathered}$$

Divide both side by -1 , we get:-

$\begin{array}{rcl}\frac{-x}{-1}& =& \frac{-2}{-1}\\ x& =& 2\end{array}$

Therefore, The value of x is 2 .