Question

problems 9 –12, solve.

1. the graph at right represents a chocolate factory during valentine week. workers packed boxes of candy over the 7 - day period.

a. represent the sequence in a table.

n	1	2	3	4	5	6	7

b. is the sequence geometric? explain.
c. write an explicit rule f(n), for the sequence of candy boxes packed. show your work to support your answer.
d. what does n and f(n) represent in context to the scenario?
e. how many total boxes of candy were packed during this time period?

Explanation:

Step1: Fill table from graph

From the right graph, extract values:
When $n=3$, $f(n)=160$; $n=4$, $f(n)=320$; $n=6$, $f(n)=1280$; $n=7$, $f(n)=2560$
Completed table:

$n$	1	2	3	4	5	6	7

Step2: Check geometric sequence

Verify ratio between terms:
$\frac{80}{40}=2$, $\frac{160}{80}=2$, $\frac{320}{160}=2$, $\frac{640}{320}=2$, $\frac{1280}{640}=2$, $\frac{2560}{1280}=2$
Common ratio $r=2$ is constant.

Step3: Derive explicit rule

Geometric sequence formula: $f(n)=a_1 r^{n-1}$
Here $a_1=40$, $r=2$, so:
$f(n)=40 \times 2^{n-1}$
Simplify: $f(n)=20 \times 2^{n}$ or $f(n)=20(2)^n$

Step4: Define variables in context

Map $n$ and $f(n)$ to the scenario.

Step5: Calculate total boxes

Sum the geometric sequence:
$S_n = a_1 \frac{r^n - 1}{r-1}$
Substitute $a_1=40$, $r=2$, $n=7$:
$S_7 = 40 \times \frac{2^7 - 1}{2-1} = 40 \times (128-1) = 40 \times 127$

Answer:

A. Completed table:

$n$	1	2	3	4	5	6	7

B. Yes, it is a geometric sequence. Each term is 2 times the previous term, so there is a constant common ratio of 2.

C. $f(n)=40 \times 2^{n-1}$ (or simplified to $f(n)=20 \times 2^n$)

D. $n$ represents the day number (from 1 to 7) during Valentine week, and $f(n)$ represents the number of candy boxes packed on day $n$.

E. 5080 total boxes