Question

What is the common ratio of the geometric sequence: $1.5$, $-9$, $54$, $-324$?

Answer 1

To solve this question we need to know about the Geometric Progression. To solve the question, the concept of common ratio should be known. Common ratio is the fraction having numerator as ${{(n+1)}^{th}}$ term and the denominator is the ${{(n)}^{th}}$mathematically it is written as $\text{common ratio=}\dfrac{{{\left( \text{n+1} \right)}^{\text{th}}}\text{term}}{{{\text{n}}^{\text{th}}}\text{term}}$.

Complete step by step solution:
The question is to find the common ratio of the numbers which are in geometric progression. The numbers in the geometric sequence are $1.5$,$-9$, $54$, $-324$. The common ratio of the terms in a sequence given could be found on dividing ${{(n+1)}^{th}}$ term by the ${{(n)}^{th}}$, which means the fraction formed will have numerator as ${{(n+1)}^{th}}$ term and the denominator is the ${{(n)}^{th}}$. On applying the same formula to the first and the second terms given in the question.
$\Rightarrow \text{common ratio=}\dfrac{{{\left( \text{n+1} \right)}^{\text{th}}}\text{term}}{{{\text{n}}^{\text{th}}}\text{term}}$
$\Rightarrow \text{common ratio=}\dfrac{{{\text{2}}^{\text{nd}}}\text{ term}}{{{1}^{\text{st}}}\text{ term}}$
On substituting the value of ${{1}^{st}}$ and ${{2}^{nd}}$ term which are $1.5$ and $-9$ respectively, we get:
$\Rightarrow \text{common ratio= }\dfrac{-9}{1.5}$
To solve it further we will have to remove the decimal sign from the denominator and the result of the multiple of $10$ will be multiplied to the numerator. In this case $10$ will be multiplied as the decimal point is after one's place of a number, $1.5$.
$\Rightarrow \text{common ratio =}\dfrac{-90}{15}$
Since the numerator is negative and the denominator is positive so the result will be negative.
$\Rightarrow \text{common ratio = -6}$
Similarly, on substituting the value of ${{2}^{nd}}$ and ${{3}^{rd}}$ term which are $-9$ and $54$ respectively in the formula of common ratio, we get:
$\Rightarrow \text{common ratio =}\dfrac{{{\text{3}}^{rd}}\text{ term}}{{{\text{2}}^{\text{nd}}}\text{term}}$
$\Rightarrow \text{common ratio =}\dfrac{54}{-9}$
Since the numerator is positive and the denominator is negative so the result will be negative.
$\Rightarrow \text{common ratio = -6}$

$\therefore$ The common ratio of the geometric sequence: $1.5$,$-9$, $54$, $-324$ is $-6$.

Note: When the common ratio of a sequence is the same then the terms are in the geometric sequence. Since the common ratio of the consecutive terms are the same, the sequence could be said to be in geometric progression. When a negative integer is multiplied or divided to the positive integer the result of the operation is negative, rest in all other cases of multiplication and division between the integers the answer is positive integer.