Question

How do you find two geometric means between $5$ and $135$ ?

Answer 1

We need to develop a geometric sequence using $5$ as the first term and $135$ as the fourth term. Then, we assume a common ratio between the consecutive terms of the sequence. Knowing the first and fourth terms, the value of the common ratio can be found out. The second and third terms are found using the formula for the ${{n}^{th}}$ term of a geometric sequence. These will be the required geometric means.

Complete step by step answer:
As we need to find the geometric mean between the two numbers, we need to assume a common ratio between the consecutive terms. Let that common ratio be $r$ .
Geometric means between the numbers $5$ and $135$ implies that if we construct a geometric sequence using the numbers $5$ and $135$ , then the terms between $5$ and $135$ will be their geometric means. Thus, for getting two geometric means between them, we construct a geometric sequence of four terms with $5$ as the first term and $135$ as the fourth term.
Now, we know that for a geometric sequence, the ${{n}^{th}}$ term can be expressed as $a{{r}^{n-1}}$ where, $a$ is the first term of the sequence and $r$ is the common ratio. In the given sequence, $5$ is the first term. This means $a=5$ . Thus, $135$ can be expressed as
$5{{r}^{3}}=135$
Dividing $5$ on both sides of the equation, we get
$\Rightarrow {{r}^{3}}=27$
Taking cube roots on both sides, we get
$\Rightarrow r=3$
Therefore, we obtain the common ratio of the required geometric sequence as $3$ . The second term is thus,
$5{{r}^{1}}$
$\Rightarrow 5\times {{3}^{1}}$
$\Rightarrow 15$
The third term is,
$5{{r}^{2}}$
$\Rightarrow 5\times {{3}^{2}}$
$\Rightarrow 45$

Therefore, we can conclude that the two geometric means between $5$ and $135$ are $15$ and $45$ respectively.

Note: Most of the students make a mistake in writing the ${{n}^{th}}$ term of a geometric sequence. Then write it as $a{{r}^{n}}$ instead of $a{{r}^{n-1}}$ . Therefore, they must be careful here as it will lead to wrong answers. We also must not try to find the geometric means by subsequent square rooting the first and the last terms.