Question

How to insert three geometric means between $6$ and $96$ ?

Answer 1

In this question, we have to insert a geometric means between two numbers. As we know, geometric mean is a special type of average, where any two consecutive terms have the same ratio. Thus, we start solving this problem by letting three different geometric means in between number 6 and 96. Then, we will find the ratio by using the formula ${{a}_{5}}=a.{{r}^{4}}$ , where a is the first term, r is the ratio and ${{a}_{5}}$ is the fifth term of the geometric mean. Then, on putting the values, we get the value of r and thus find the three ratios between the numbers. On further simplification, we get the required solution for the problem.

Complete step-by-step solution:
According to the question, we have to find the value of geometric means between two numbers.
Let us suppose the geometric mean after inserting three numbers between them are $6,{{a}_{2}},{{a}_{3}},{{a}_{4}},96$ --- (1)
Now, we will find the ratio of the geometric mean by using the formula ${{a}_{5}}=a.{{r}^{4}}$ --- (2)
As we know, from the equation (1), we get $a=6,$ ${{a}_{5}}=96,$ and $r=r$ ----- (3)
Therefore, we put the value of equation (3) in equation (2), we get
$96=(6).{{r}^{4}}$
Now, we will divide 6 on both sides in the above equation, we get
$\dfrac{96}{6}=\dfrac{6}{6}.{{r}^{4}}$
On further simplification, we get
$16={{r}^{4}}$
Now, we will take the power $\frac{1}{4}$ on both sides in the above equation, we get
${{\left( 16 \right)}^{\dfrac{1}{4}}}={{\left( {{r}^{4}} \right)}^{\dfrac{1}{4}}}$
On further solving, we get
$r=2$
Thus, the ratio of the geometric mean is equal to 2. Hence, the value of the three geometric means, we get
$\begin{aligned} & {{a}_{2}}=6.{{r}^{1}} \\\ & {{a}_{2}}=6.(2) \\\ \end{aligned}$
Therefore, we get
${{a}_{2}}=12$
Also, for the next geometric mean
$\begin{aligned} & {{a}_{3}}=6.{{r}^{2}} \\\ & {{a}_{3}}=6.{{(2)}^{2}} \\\ \end{aligned}$
Therefore, we get
${{a}_{3}}=24$
Thus, for the next geometric mean
$\begin{aligned} & {{a}_{3}}=6.{{r}^{3}} \\\ & {{a}_{3}}=6.{{(2)}^{3}} \\\ \end{aligned}$
Therefore, we get
${{a}_{3}}=48$
Therefore, for the geometric mean between two numbers 6 and 96, its values are $12,24,48$ .

Note: While solving this problem, keep in mind the definition of the geometric mean. Do step-by-step calculations to avoid confusion and mathematical error. Remember when we add 3 numbers between two numbers, we get a total of 5 numbers and not 4 numbers.