Question

write an explicit formula for $a_n$, the $n^{th}$ term of the sequence $6, -24, 96,....$

Explanation:

Step1: Identify sequence type

This is a geometric sequence (ratio between consecutive terms is constant).

Step2: Find first term $a_1$

The first term $a_1 = 6$

Step3: Calculate common ratio $r$

$r = \frac{-24}{6} = -4$, verify with $\frac{96}{-24} = -4$

Step4: Apply geometric sequence formula

The explicit formula for a geometric sequence is $a_n = a_1 \cdot r^{n-1}$
Substitute $a_1=6$ and $r=-4$:
$a_n = 6 \cdot (-4)^{n-1}$

Answer:

$a_n = 6 \cdot (-4)^{n-1}$