Question

For what values of \[x\] , the number \[ - \dfrac{2}{7}x,x, - \dfrac{7}{2}\] are in G.P?

Answer 1

Hint:In order to solve the problem related to Geometric progression; use the concept of geometric mean of the series. With the help of formulas for geometric mean, find some equation from the problem statement in terms of the given variable x. Further solve the quadratic equation obtained and select the practical root as the value of x from the set of two roots that will be obtained.

Complete step-by-step answer:
Given that:
\[ - \dfrac{2}{7}x,x, - \dfrac{7}{2}\] are in G.P
As we know that of the terms a, b and c are in geometric progression then the middle term b is the geometric mean of the given series and also we have the relationship between the three terms in geometric progression which is given as:
\[{b^2} = ac\]
Using the above formula for the given geometric progression series, we have:
\[
  \because {b^2} = ac \\
   \Rightarrow {x^2} = \left( {\dfrac{{ - 2}}{7}x} \right) \times \dfrac{{ - 7}}{2} \\
 \]
Let us arrange the above equation and bring the terms on one side.
\[
   \Rightarrow {x^2} = x \\
   \Rightarrow {x^2} - x = 0 \\
 \]
The above equation is a quadratic equation in terms of variable x. Let us solve the quadratic equation directly by taking the common term in order to find the value of x.
\[
  \because {x^2} - x = 0 \\
   \Rightarrow x\left( {x - 1} \right) = 0 \\
   \Rightarrow x = 0{\text{ or }}\left( {x - 1} \right) = 0 \\
   \Rightarrow x = 0{\text{ or }}x = 1 \\
 \]
The roots of quadratic equations are 0 and 1.
When we substitute the value of x as 0 in the geometric series the series becomes (0, 0 and 0) which is not a valid geometric progression.
So value "0" of the variable x is not a practical answer.
So the root of x is 1 from the above equation.
Hence, the value of x is 1.

Note:In order to solve such problems related to series, students must remember the formula for relationship between the consecutive terms of the series may it be arithmetic progression series or geometric progression series. Also students after solving the quadratic equation in such types of problems must only consider the practically feasible roots as some of the roots obtained might not be practically feasible.