Answer: The correct option is (b) Yes, [tex]\dfrac{3}{4}.[/tex]
Step-by-step explanation: The given sequence is
[tex]\dfrac{1}{4},~\dfrac{3}{16},~\dfrac{9}{64},~\dfrac{27}{256},~\dfrac{81}{1024},~.~.~.[/tex]
We are to check whether the given sequence is geometric. If so, we are to find the common ration.
GEOMETRIC SEQUENCE: A sequence of numbers where each term is found by multiplying by a constant with the preceding term. This constant is called the common ratio, r.
In the given sequence, we can see
[tex]\dfrac{\frac{3}{16}}{\frac{1}{4}}=\dfrac{3}{4}~~~~\Rightarrow \dfrac{3}{16}=\dfrac{3}{4}\times \dfrac{1}{4},\\\\\\\dfrac{\frac{9}{64}}{\frac{3}{16}}=\dfrac{3}{4}~~~~\Rightarrow \dfrac{9}{64}=\dfrac{3}{4}\times \dfrac{3}{16},\\\\\\\dfrac{\frac{27}{256}}{\frac{9}{64}}=\dfrac{3}{4}~~~~\Rightarrow \dfrac{27}{256}=\dfrac{3}{4}\times \dfrac{9}{64},\\\\\\\dfrac{\frac{81}{1024}}{\frac{27}{256}}=\dfrac{3}{4}~~~~\Rightarrow \dfrac{81}{1024}=\dfrac{3}{4}\times \dfrac{27}{256},~etc[/tex]
Therefore, each term is found by multiplying [tex]\dfrac{3}{4}[/tex] with the preceding term.
Thus, the given sequence is geometric and its common ratio is [tex]\dfrac{3}{4}[/tex]
Option (b) is correct.
