Question

1. a portion of a geometric sequence is shown in the table. a. complete the table. (2 points) b. write the explicit rule for the sequence. (2 points) term number, n term a_n 1 2 3 162 4 486 5 1458

Explanation:

Step1: Find the common ratio $r$

For a geometric sequence, $r=\frac{a_{n + 1}}{a_{n}}$. Using $n = 3$ and $n=4$, $r=\frac{486}{162}=3$.

Step2: Find the first - term $a_1$

The formula for the $n$th term of a geometric sequence is $a_{n}=a_{1}r^{n - 1}$. When $n = 3$, $a_{3}=a_{1}r^{2}$. Substituting $a_{3}=162$ and $r = 3$ into $a_{3}=a_{1}r^{2}$, we get $162=a_{1}\times3^{2}$. Then $162 = 9a_{1}$, so $a_{1}=\frac{162}{9}=18$.

Step3: Write the explicit rule

The explicit rule for a geometric sequence is $a_{n}=a_{1}r^{n - 1}$. Substituting $a_{1}=18$ and $r = 3$ into the formula, we have $a_{n}=18\times3^{n - 1}$.

Answer:

$a_{n}=18\times3^{n - 1}$