Question

3, -12, 48, -192, ... write the explicit formula for the geometric sequence. \\(a_n = 3(4)^{n - 1}\\) \\(a_n = 4(3)^{n - 1}\\) \\(a_n = -4(3)^{n - 1}\\) \\(a_n = 3(-4)^{n - 1}\\)

Explanation:

Step1: Identify first term $a_1$

The first term of the sequence is $a_1 = 3$.

Step2: Calculate common ratio $r$

Find ratio of term 2 to term 1:
$r = \frac{-12}{3} = -4$
Verify with term 3 to term 2: $\frac{48}{-12} = -4$, which is consistent.

Step3: Apply geometric sequence formula

The explicit formula for a geometric sequence is $a_n = a_1(r)^{n-1}$. Substitute $a_1=3$ and $r=-4$.
$a_n = 3(-4)^{n-1}$

Answer:

D. $a_n = 3(-4)^{n-1}$