Question

For what values of x, the numbers \[-1,x,-\dfrac{3}{4}\] are in GP?
1. \[\dfrac{3}{2}\]
2. \[\dfrac{\sqrt{3}}{4}\]
3. \[\dfrac{\sqrt{3}}{2}\]
4. \[\dfrac{3}{\sqrt{2}}\]

Answer 1

Hint: To solve this question one must know what a geometric progression and series is. A geometric progression is a progression of numbers; this progression is done with there being a constant ratio between each number and the number previous to it. As from the information written above we can express the first three numbers of any geometric progression. Those numbers being \[a,ar,a{{r}^{2}}\]. Now you can further use this information of the first three numbers of any geometric progression to find the value of x.

Complete step-by-step answer:
Now we know the first term of any geometric progression is represented by the letter ‘a’ and the constant ratio or common ratio is denoted by the letter ‘r’. We have three numbers which are in a geometric ratio which are \[-1,x,\dfrac{-3}{4}\]
From this we can see that the first term of the geometric progression is \[-1\], hence we have;
\[a=-1\]
Let the common ratio be r, then now we know using that we can find the second and third term will be given by \[ar,a{{r}^{2}}\]. Hence through this we get;
Now we know and can see that
\[ar=x\]
And also;
\[a{{r}^{2}}=-\dfrac{3}{4}\]
Now after this we can solve this using the formula \[{{b}^{2}}=ac\] .
Here a is the first term here, b will be the second term and c will be the third term in the progression.
Putting the values here we get
\[{{x}^{2}}=-1\times -\dfrac{3}{4}\]
Now multiplying and taking the root we get the answer of x which will be \[x=\dfrac{\sqrt{3}}{2},-\dfrac{\sqrt{3}}{2}\]. Here in this question the answer will be option 3
Alternatively;
We can put the value of a in the third term which will give us
\[(-1){{r}^{2}}=-\dfrac{3}{4}\]
Now we will get r as \[r=\dfrac{\sqrt{3}}{2},-\dfrac{\sqrt{3}}{2}\]
Multiplying this we will get the value of x which is the second term represented by ar
\[x=ar=\dfrac{\sqrt{3}}{2},-\dfrac{\sqrt{3}}{2}\]

Note: There are two solutions of x both of them are important and give the needed answer in this question since we have to select the answer from the option that’s the only reason only one of them is in the answer but otherwise both of the answers need to be mentioned or else it might lead to your answer not getting full credits.