Question

17
given $a_4 = 5103$ and that the common ratio is 9, write the explicit rule for the sequence. (a.fgr.9.4)
a $a_n = -9(7)^{n-1}$
b $a_n = 7(9)^{n-1}$
c $a_n = 9(7)^{n-1}$
d $a_n = -7(9)^{n-1}$

Explanation:

Step1: Recall geometric sequence formula

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

Step2: Plug known values into formula

We know $a_4 = 5103$, $r = 9$, and $n=4$. Substitute into the formula:
$5103 = a_1(9)^{4-1}$

Step3: Simplify the exponent

Calculate $9^3$:
$9^3 = 729$, so $5103 = a_1(729)$

Step4: Solve for $a_1$

Rearrange to solve for $a_1$:
$a_1 = \frac{5103}{729} = 7$

Step5: Write the explicit rule

Substitute $a_1=7$ and $r=9$ back into the general formula:
$a_n = 7(9)^{n-1}$

Answer:

B. $a_n = 7(9)^{n-1}$