Answer:
Option B) [tex]a_{n} = 2\cdot 4^{n-1}[/tex]
Step-by-step explanation:
The given geometric sequence is
2, 8, 32, 128,....
The general form of a geometric sequence is given by
[tex]a_{n} = a_{1}\cdot r^{n-1}[/tex]
Where n is the nth term that we want to find out.
a₁ is the first term in the geometric sequence that is 2
r is the common ratio and can found by simply dividing any two consecutive numbers in the sequence,
[tex]r=\frac{8}{2} = 4[/tex]
You can try other consecutive numbers too, you will get the same common ratio
[tex]r=\frac{32}{8} = 4[/tex]
[tex]r=\frac{128}{32} = 4[/tex]
So the common ratio is 4 in this case.
Substitute the value of a₁ and r into the above general equation
[tex]a_{n} = 2\cdot 4^{n-1}[/tex]
This is the general form of the given geometric sequence.
Therefore, the correct option is B
Note: Don't multiply the first term and common ratio otherwise you wont get correct results.
Verification:
[tex]a_{n} = 2\cdot 4^{n-1}[/tex]
Lets find out the 2nd term
Substitute n = 2
[tex]a_{2} = 2\cdot 4^{2-1} = 2\cdot 4^{1} = 2\cdot 4 = 8[/tex]
Lets find out the 3rd term
Substitute n = 3
[tex]a_{3} = 2\cdot 4^{3-1} = 2\cdot 4^{2} = 2\cdot 16 = 32[/tex]
Lets find out the 4th term
Substitute n = 4
[tex]a_{4} = 2\cdot 4^{4-1} = 2\cdot 4^{3} = 2\cdot 64 = 128[/tex]
Lets find out the 5th term
Substitute n = 5
[tex]a_{5} = 2\cdot 4^{5-1} = 2\cdot 4^{4} = 2\cdot 256 = 512[/tex]
Hence, we are getting correct results!
