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

Hint: 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.