Question

1. a portion of a geometric sequence is shown in the table.

a. complete the table. (2 points)

term number, n	1	2	3	4	5
term (a_{n})			162	486	1458

b. write the explicit rule for the sequence. (2 points)

1. a linear relationship exists between (x) and (y) that can be written as (4x + 15y=c), where (c) is a constant. the table gives some values of (x) and their corresponding values of (y). find the value of (c). (2 points)

(x)	-12	0	15	52.5
(y)	17.2	14	10	0

1. write an equation for the linear relationship shown on the graph. (2 points)

Explanation:

7.

a.

Step1: Find the common ratio

For a geometric sequence, the common ratio $r$ is found by dividing a term by its previous term. $r=\frac{486}{162} = 3$.

Step2: Find the first - term

Let the first - term be $a_1$. We know that $a_n=a_1r^{n - 1}$. Using $n = 3$, $a_3=a_1r^{2}$, and since $a_3 = 162$ and $r = 3$, we have $162=a_1\times3^{2}$, so $a_1=\frac{162}{9}=18$.

Step3: Fill in the table

$a_2=a_1r=18\times3 = 54$.
The completed table:

Term Number, $n$	1	2	3	4	5

The explicit rule for a geometric sequence is $a_n=a_1r^{n - 1}$. Since $a_1 = 18$ and $r = 3$, the explicit rule is $a_n=18\times3^{n - 1}$.

Step1: Use the equation $4x + 15y=C$

We can take any pair of $(x,y)$ values from the table. Let's take $(x = 0,y = 14)$. Substitute into the equation $4x+15y=C$.

Step2: Calculate $C$

When $x = 0$ and $y = 14$, we have $4\times0+15\times14=C$, so $C = 210$.

Answer:

Term Number, $n$	1	2	3	4	5

b.