Question

The product of five positive numbers in GP is $32$ , and the ratio of the greatest number to the smallest number is $81:1.$Find the numbers.

Answer 1

The given series is in the Geometrical Progression form as each consecutive term is multiplied by a fixed ratio. A Geometrical Progression is a sequence of numbers where each term is multiplied by its previous number of sequences with a constant number known as a common ratio$r$. In general, a common ratio $r$ is found by dividing any term of the series with its previous term. The behaviour of a geometric series depends on its common ratio.
The sum of a Geometrical decreasing series is given as${S_n} = \dfrac{a}{{1 - r}};r < 1$, whereas for a Geometrical increasing series ${S_n} = \dfrac{a}{{r - 1}};r > 1$

Here we will be assuming the numbers as $\dfrac{a}{{{r^2}}},\dfrac{a}{r},a,ar,a{r^2}$ and find the first term and common ratio, and thereby find the other terms.

Complete step-by-step solution:
The product of five positive numbers in GP is 32, and the ratio of the greatest number to the smallest number is 81:1.

Let us assume that the five numbers in GP are in the form of $\dfrac{a}{{{r^2}}},\dfrac{a}{r},a,ar,a{r^2}$
Then their product

$$= \dfrac{a}{{{r^2}}}.\dfrac{a}{r}.a.ar.a{r^2} \\\ = {a^5} \\\$$

Since the product is given as 32, so

$${a^5} = 32 \\\ {a^5} = {2^5} \\\ a = 2 \\\$$

Again, since the ratio of the greatest number to the smallest number is 81:1,

$$\dfrac{{a{r^2}}}{{\left( {\dfrac{a}{{{r^2}}}} \right)}} = 81 \\\ {r^4} = 81 \\\ {r^4} = {3^4} \\\ r = 3 \\\$$

Hence the numbers are:
$\dfrac{a}{{{r^2}}},\dfrac{a}{r},a,ar,a{r^2} \to \dfrac{2}{9},\dfrac{2}{3},2,6,18$

Additional Information: If the ratio,
$r = 1$The progression is constant; all the terms in the series are the same.
$r > 1$The progression is increasing; all the subsequent terms in the series are increasing by the common factor.
$r < 1$, the progression is decreasing; all the subsequent terms in the series are decreasing by the common factor.
Mathematically, a geometric progression series is summarized as ${a_1},{a_1}r,{a_1}{r^2},{a_1}{r^3}........$where ${a_1}$ is the first term of series and $r$ is the common ratio.

Note: In these types of questions, it is to be always remembered that assuming the numbers as $\dfrac{a}{{{r^2}}},\dfrac{a}{r},a,ar,a{r^2}$saves calculation since the common ratio gets cancelled whereas if we would have chosen in the traditional way we would have to do complex calculations.