Question

How do you determine whether the sequence $\dfrac{1}{8},\dfrac{1}{4},\dfrac{1}{2},1,\cdots \cdots $ is geometric and if it is what is the common ratio?

Answer 1

Hint: In this question, we are given a sequence and we need to check if it is a geometric sequence or not. We need to find the common ratio if the sequence is geometric. For this, we will just check the ratio of numbers. If there exists a common ratio between the numbers in a sequence then this implies that this sequence is a geometric sequence. We will take ratio as $\dfrac{\text{Second term}}{\text{First term}},\dfrac{\text{Third term}}{\text{Second term}}$.

Complete step by step answer:
Here we are given the number in sequence as $\dfrac{1}{8},\dfrac{1}{4},\dfrac{1}{2},1,\cdots \cdots $.
We need to check if this sequence is a geometric sequence or not.
As we know that, a sequence is geometric if it has the same common ratio between each term. So we will use the concept here.
Let the common ratio be r. Common ratio is the number you multiply it by a number and get the next number of the sequence.
For the first two terms we should have $\dfrac{1}{8}\times r=\dfrac{1}{4}$.
For second and third terms we should have $\dfrac{1}{4}\times r=\dfrac{1}{2}$.
Let us calculate the common ratio in each case. For the first two terms $\dfrac{1}{8}\text{ and }\dfrac{1}{4}$. The ratio will be taken as $\dfrac{\text{Second term}}{\text{First term}}$. We get $r=\dfrac{\dfrac{1}{4}}{\dfrac{1}{8}}$.
Simplifying the fractional part we get $r=\dfrac{8}{4}$.
When 8 is divided by 4 we get 2. So the ratio becomes r = 2.
Now let us calculate the ratio for the second term and the third term. The ratio will be taken as $\dfrac{\text{Third term}}{\text{Second term}}$ we get $r=\dfrac{\dfrac{1}{2}}{\dfrac{1}{4}}$.
Simplifying the fractional part we get $r=\dfrac{4}{2}$.
When 4 is divided by 2 we get 2. So the ratio becomes r = 2.
As we can see, the common ratio is the same. So the sequence is the geometric sequence.
We have already calculated the common ratio which is equal to 2.
Hence the given sequence is a geometric sequence with the common ratio as 2.

Note:
Students should take care while taking care. Note that the next term is taken in the numerator. Students can calculate the ratio for the next two numbers as well to make the answer more clear. The term in a geometric series are written in the form as a, ar, $a{{r}^{2}},a{{r}^{3}}\ldots \ldots $ where r is the common ratio.