Question

A geometric sequence is shown below.

−12,−2,−8,−32,...
What is an explicit representation for the nth term of the sequence?
A.f(n)=12(−4)n
B. f(n)=−12(4)n−1
C. f(n)=−12(4)n
D. f(n)=12(−4)n−1

Answer 1

An explicit representation for the nth term of the sequence:

`f_n=-(1)/(2)\cdot \:4^(n-1)`

It means, option (B) should be true.

Explanation:

Given the geometric sequence

`-(1)/(2),\:-2,\:-8,\:-32,...`

A geometric sequence has a constant ratio, denoted by 'r', and is defined by

`f_n=f_1\cdot r^(n-1)`

Determining the common ratios of all the adjacent terms

`(-2)/(-(1)/(2))=4,\:\quad (-8)/(-2)=4,\:\quad (-32)/(-8)=4`

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

`f_n=f_1\cdot r^(n-1)`

`f_n=-(1)/(2)\cdot \:4^(n-1)`

Thus, an explicit representation for the nth term of the sequence:

`f_n=-(1)/(2)\cdot \:4^(n-1)`

It means, option (B) should be true.