Question

value 3
what is the explicit formula of the geometric sequence?
2, -8, 32, -128,
a. $a_n = -128(-4)^n$
b. $a_n = -4(2)^{n - 1}$
c. $a_n = 2(-8)^{n - 1}$
d. $a_n = 2(-4)^{n - 1}$

Explanation:

Step1: Identify first term $a_1$

The first term of the sequence is $a_1=2$.

Step2: Calculate common ratio $r$

Find ratio of term 2 to term1:
$r=\frac{-8}{2}=-4$
Verify with term3/term2: $\frac{32}{-8}=-4$, consistent.

Step3: Apply geometric sequence formula

The explicit formula for a geometric sequence is $a_n=a_1(r)^{n-1}$. Substitute $a_1=2$ and $r=-4$:
$a_n=2(-4)^{n-1}$

Step4: Verify with given options

Match the derived formula to the options.

Answer:

d. $a_n=2(-4)^{n-1}$