Question

How do you find two geometric means between 7 and 875 ?

Answer 1

Hint: To two geometric means between 7 and 875, we will consider a geometric series $a,ar,a{{r}^{2}},a{{r}^{3}}$ , where a is the first term and r is the common ratio. We will represent $a=7$ and $a{{r}^{3}}=875$ . $ar$ and $a{{r}^{2}}$ will be the required geometric means. From $a{{r}^{3}}=875$ , we can get the value of r. Substituting the value of a and r in $ar$ and $a{{r}^{2}}$ , we will get the required answer.

Complete step by step solution:
We need to find two geometric means between 7 and 875. Let us consider a geometric series $a,ar,a{{r}^{2}},a{{r}^{3}}$ , where a is the first term and r is the common ratio. We will represent $a=7$ and $a{{r}^{3}}=875$ . We have to find the two geometric means between 7 and 875, that is, $ar$ and $a{{r}^{2}}$ .
Now, let us consider $a{{r}^{3}}=875$ . We can find r from this.
$\begin{align}
  & a{{r}^{3}}=875 \\
 & \Rightarrow 7{{r}^{3}}=875 \\
\end{align}$
Let us take r from LHS to RHS.
$\Rightarrow {{r}^{3}}=\dfrac{875}{7}=125$
Let us take the cube root.
$\Rightarrow r=\sqrt[3]{125}=5$
Now, we can find $ar$ and $a{{r}^{2}}$ by substituting the value of a and r.
$\Rightarrow ar=7\times 5=35$
$\Rightarrow a{{r}^{2}}=7\times {{\left( 5 \right)}^{2}}=7\times 25=175$

Hence, the two geometric means between 7 and 875 are 35 and 175.

Note: Students must be aware that when these types of questions are asked, do not use the formula of geometric mean of a series, that is, \[GM=\sqrt[n]{{{x}_{1}},{{x}_{2}},...,{{x}_{n}}}\] , where \[{{x}_{1}},{{x}_{2}},...,{{x}_{n}}\] are the observations. When we have to find the three geometric means between two numbers, we will write the series as $a,ar,a{{r}^{2}},a{{r}^{3}},a{{r}^{4}}$ . In general, if we have to find n geometric means between 2 numbers, we will write the series with $\left( n+2 \right)$ terms.