Question

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

Explanation:

Step1: Identify first term $a_1$

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

Step2: Calculate common ratio $r$

Divide any term by the prior term: $r = \frac{8}{2} = 4$.

Step3: Use 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 terms

For $n=2$: $a_2=2(4)^{2-1}=2(4)=8$, which matches. For $n=3$: $a_3=2(4)^{3-1}=2(16)=32$, which matches.

Answer:

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