Answer:
An explicit representation for the nth term of the sequence:
[tex]f_n=-\frac{1}{2}\cdot \:4^{n-1}[/tex]
It means, option (B) should be true.
Step-by-step explanation:
Given the geometric sequence
[tex]-\frac{1}{2},\:-2,\:-8,\:-32,...[/tex]
A geometric sequence has a constant ratio, denoted by 'r', and is defined by
[tex]f_n=f_1\cdot r^{n-1}[/tex]
Determining the common ratios of all the adjacent terms
[tex]\frac{-2}{-\frac{1}{2}}=4,\:\quad \frac{-8}{-2}=4,\:\quad \frac{-32}{-8}=4[/tex]
As the ratio is the same, so
r = 4
Given that f₁ = -1/2
substituting r = 4, and f₁ = -1/2 in the nth term
[tex]f_n=f_1\cdot r^{n-1}[/tex]
[tex]f_n=-\frac{1}{2}\cdot \:4^{n-1}[/tex]
Thus, an explicit representation for the nth term of the sequence:
[tex]f_n=-\frac{1}{2}\cdot \:4^{n-1}[/tex]
It means, option (B) should be true.
