Question

Find an equation for the nth term of a geometric sequence where the second and fifth terms are -8 and 512, respectively.

Answer 1

The correct answer is C) $a_n=2\times(-4)^{n-1}$.

Since it is a geometric sequence, we multiply by a constant, r, each time to find the next term.

g₂ = -8
g₅ = 512

g₂ * r * r * r = g₅
-8(r)(r)(r) = 512
-8r³ = 512

Divide both sides by -8:
-8r³/-8 = 512/-8
r³ = -64

Take the cubed root of both sides:
∛r³ = ∛-64
r = -4

Now we work backward from g₂ to find g₁:

-8/-4 = 2

We have that g₁ = 2 and r = -4.  This gives us

$a_n=g_1 \times r^{n-1} \\ \\a_n=2 \times (-4)^{n-1}$