Question

the first four terms of a geometric sequence are shown.
\\(\frac{1}{16}, \frac{1}{4}, 1, 4, \dots\\)
which is the explicit formula for the sequence?
\\(\circ\\) \\(a_n = \frac{1}{16} \cdot 4^{n - 1}\\)
\\(\circ\\) \\(a_n = 4 \cdot \left(\frac{1}{16}\
ight)^{n - 1}\\)
\\(\circ\\) \\(a_n = 4 \cdot \left(\frac{1}{64}\
ight)^{n - 1}\\)
\\(\circ\\) \\(a_n = \frac{1}{64} \cdot 4^{n - 1}\\)

Explanation:

Step1: Recall geometric sequence formula

The explicit formula for a geometric sequence is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term, and $r$ is the common ratio.

Step2: Identify $a_1$

From the sequence, the first term $a_1 = \frac{1}{16}$.

Step3: Calculate common ratio $r$

Find $r$ by dividing the second term by the first term: $r = \frac{\frac{1}{4}}{\frac{1}{16}} = \frac{1}{4} \times 16 = 4$.

Step4: Substitute into formula

Substitute $a_1 = \frac{1}{16}$ and $r=4$ into the explicit formula: $a_n = \frac{1}{16} \cdot 4^{n-1}$.

Answer:

$a_n = \frac{1}{16} \cdot 4^{n-1}$