Question

What is the common ratio of geometric sequence 1.1, 4.4, 17.6, 70.4, …….?

Answer 1

For solving this question you should know about the geometric progression. To solve the question, the concept of common ratio should be common. Common ratio is the fraction having numerator as ${{\left( n+1 \right)}^{th}}$ term and the denominator is the ${{\left( n \right)}^{th}}$ term mathematically it is written as common ratio $=\dfrac{{{\left( n+1 \right)}^{th}}term}{{{n}^{th}}term}$.

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 1.1, 4.4, 17.6, 70.4. The common ratio of the terms in a sequence given could be found on dividing ${{\left( n+1 \right)}^{th}}$ term by the ${{\left( n \right)}^{th}}$ term, which means the fraction formed will have numerator as ${{\left( n+1 \right)}^{th}}$ term and the denominator as ${{\left( n \right)}^{th}}$ term. On applying the same formula to the first and second terms given in the question:
$\Rightarrow$ Common ratio $=\dfrac{{{\left( n+1 \right)}^{th}}term}{{{n}^{th}}term}$
$\Rightarrow$ Common ratio $=\dfrac{{{2}^{nd}}term}{{{1}^{st}}term}$
On substituting the value of ${{1}^{st}}$ and ${{2}^{nd}}$ term which are 1.1 and 4.4 respectively, we get,
$\Rightarrow$ Common ratio $=\dfrac{4.4}{1.1}$
To solve this we have to remove the decimal signs of both numerator and denominator. And multiply this by $\dfrac{10}{10}$. In this case the fraction will be a whole number fraction and if we don’t remove it to the decimal then too we can solve it by just dividing both.
$\Rightarrow$ Common ratio $=\dfrac{4.4}{1.1}\times \dfrac{10}{10}=\dfrac{44}{11}=4$
Similarly, on substituting the values of ${{2}^{nd}}$ and ${{3}^{rd}}$ term which are 4.4 and 17.6 respectively in the formula of common ratio, we get,
$\Rightarrow$ Common ratio $=\dfrac{{{3}^{rd}}term}{{{2}^{nd}}term}$
$\Rightarrow$ Common ratio $=\dfrac{17.6}{4.4}=4$
And for ${{3}^{rd}}$ and ${{4}^{th}}$ term
$\Rightarrow$ Common ratio $=\dfrac{{{4}^{th}}term}{{{3}^{rd}}term}$
$\Rightarrow$ Common ratio $=\dfrac{70.4}{17.6}=4$

So, the common ratio of G.B. 1.1, 4.4, 17.6, 70.4 is the 4.

Note:
When the common ratio of the sequence is the same then the terms are in geometric sequence. Since, the common ratio of the consecutive could be said to be in geometric progression.