3. in a larger rug, atsidi plans a repeating geometric design where each new section doubles in size. if the first section uses 50 threads and the pattern continues for 8 sections:
a. calculate the total number of threads used after 8 sections.
b. what could be a rule or formula ( either explicit or recursive) for the total number of threads for n sections.
c. explain in your own words how this relates to exponential growth.
4 atsidi wants to create a border pattern that increases linearly. the first border has 6 colored triangles, the second 9, the third 12, and so on. she wants to know how many total triangles she will have after finishing 20 borders.
a. find the total number of triangles used in 20 borders.
b. what could be a rule or formula ( explicit or recursive) for the sum of triangles of n borders.
c. how can atsidi use this information to estimate how many yarn bundles to prepare?

Question

Accepted Answer

### Part 3a:
# Explanation:
## Step1: Identify the sequence type
The number of threads per section forms a geometric sequence with first term $ a_1 = 50 $ and common ratio $ r = 2 $ (since each section doubles in size). The formula for the sum of the first $ n $ terms of a geometric sequence is $ S_n=\frac{a_1(1 - r^n)}{1 - r} $ (for $ r
eq1 $).
## Step2: Substitute values for $ n = 8 $
Here, $ a_1 = 50 $, $ r = 2 $, and $ n = 8 $. Substitute into the formula:
$$
\begin{align*}
S_8&=\frac{50(1 - 2^8)}{1 - 2}\
&=\frac{50(1 - 256)}{-1}\
&=\frac{50(-255)}{-1}\
&=50\times255\
& = 12750
\end{align*}
$$
# Answer:
12750

### Part 3b:
# Explanation:
## Step1: Explicit formula for sum of geometric sequence
For a geometric sequence with first term $ a_1 $ and common ratio $ r $, the explicit formula for the sum of the first $ n $ terms is $ S_n=\frac{a_1(1 - r^n)}{1 - r} $ (when $ r
eq1 $). Here, $ a_1 = 50 $ and $ r = 2 $, so substituting these values gives $ S_n=\frac{50(1 - 2^n)}{1 - 2} $. Simplifying the denominator ($ 1 - 2=-1 $), we get $ S_n = 50(2^n - 1) $ (since dividing by - 1 flips the signs in the numerator).
## Step2: Recursive formula
The recursive formula for the sum of the first $ n $ terms of a geometric sequence can be defined in terms of the previous sum. Let $ S_n $ be the sum of the first $ n $ terms and $ S_{n - 1} $ be the sum of the first $ n - 1 $ terms. The $ n $-th term $ a_n=a_1r^{n - 1}=50\times2^{n - 1} $. So, $ S_n=S_{n - 1}+a_n=S_{n - 1}+50\times2^{n - 1} $, with the initial condition $ S_1 = 50 $ (since the first section uses 50 threads).
# Answer:
Explicit formula: $ S_n = 50(2^n - 1) $; Recursive formula: $ S_n=S_{n - 1}+50\times2^{n - 1} $, $ S_1 = 50 $

### Part 3c:
# Explanation:
## Step1: Definition of exponential growth
Exponential growth occurs when a quantity increases by a constant factor over equal intervals of time (or in this case, over equal numbers of sections). In this problem, the number of threads per section is doubling (multiplying by 2, a constant factor) with each new section. The total number of threads used after $ n $ sections is given by a formula that involves $ 2^n $ (from the geometric sequence sum formula), which is the characteristic of exponential growth. As $ n $ increases, the total number of threads grows very rapidly because the base of the exponential term (2) is greater than 1, leading to a rapid increase in the value of the function as $ n $ increases.
## Step2: Relate to the problem
The number of threads per section is a geometric sequence with a common ratio greater than 1 ( $ r = 2 $ ), so the total number of threads (the sum of the geometric sequence) will grow exponentially. For example, as we saw in part 3a, after 8 sections, the total is 12750, and if we were to calculate for a larger $ n $, say $ n = 10 $, the total would be $ 50(2^{10}-1)=50(1024 - 1)=50\times1023 = 51150 $, which is a much larger increase compared to linear growth.
# Answer:
This relates to exponential growth because the number of threads per section forms a geometric sequence with a common ratio ($ r = 2 $) greater than 1. The total number of threads (sum of the geometric sequence) is given by a formula involving $ 2^n $, which means the total number of threads grows rapidly (exponentially) as the number of sections ($ n $) increases, with each section contributing a number of threads that is double the previous section's count.

### Part 4a:
# Explanation:
## Step1: Identify the sequence type
The number of triangles per border forms an arithmetic sequence with first term $ a_1 = 6 $, common difference $ d=9 - 6 = 3 $ (since $ 9-6 = 3 $ and $ 12 - 9=3 $). The formula for the sum of the first $ n $ terms of an arithmetic sequence is $ S_n=\frac{n}{2}[2a_1+(n - 1)d] $.
## Step2: Substitute values for $ n = 20 $
Here, $ n = 20 $, $ a_1 = 6 $, and $ d = 3 $. Substitute into the formula:
$$
\begin{align*}
S_{20}&=\frac{20}{2}[2\times6+(20 - 1)\times3]\
&=10[12 + 19\times3]\
&=10[12+57]\
&=10\times69\
&=690
\end{align*}
$$
# Answer:
690

### Part 4b:
# Explanation:
## Step1: Explicit formula for sum of arithmetic sequence
The formula for the sum of the first $ n $ terms of an arithmetic sequence is $ S_n=\frac{n}{2}[2a_1+(n - 1)d] $. Here, $ a_1 = 6 $ and $ d = 3 $, so substituting these values gives $ S_n=\frac{n}{2}[2\times6+(n - 1)\times3] $. Simplifying the expression inside the brackets: $ 12+3n - 3=3n + 9 $. So, $ S_n=\frac{n}{2}(3n + 9)=\frac{3n(n + 3)}{2} $ (or we can also write it as $ S_n=3n+\frac{3n(n - 1)}{2} $ which is the recursive - like expansion, but the explicit form is more straightforward here).
## Step2: Recursive formula
For an arithmetic sequence, the recursive formula for the sum can be defined as $ S_n=S_{n - 1}+a_n $, where $ a_n=a_1+(n - 1)d $. Here, $ a_1 = 6 $ and $ d = 3 $, so $ a_n=6+(n - 1)\times3=3n + 3 $. And $ S_1=a_1 = 6 $, so the recursive formula is $ S_n=S_{n - 1}+3n + 3 $ with $ S_1 = 6 $.
# Answer:
Explicit formula: $ S_n=\frac{n}{2}[2\times6+(n - 1)\times3] $ (or simplified $ S_n=\frac{3n(n + 3)}{2} $); Recursive formula: $ S_n=S_{n - 1}+3n + 3 $, $ S_1 = 6 $

### Part 4c:
# Explanation:
## Step1: Relate triangles to yarn bundles
First, Atsidi can determine the relationship between the number of triangles and the amount of yarn required per triangle (let's say, if she knows how much yarn is needed to make one triangle, she can call this amount $ y $ (in some unit, like length of yarn per triangle)). Then, the total amount of yarn needed for $ N $ triangles is $ Y = N\times y $.
## Step2: Use the sum formula
From part 4a, she found the total number of triangles for 20 borders. If she wants to estimate for a different number of borders (say $ n $ borders), she can use the formula from part 4b (either the explicit or recursive formula) to find the total number of triangles for that $ n $. Then, multiply the total number of triangles by the amount of yarn per triangle to get the total yarn needed. This will help her decide how many yarn bundles to prepare, considering the size of each yarn bundle (if each bundle has a certain amount of yarn, say $ B $ units, then the number of bundles $ k=\lceil\frac{Y}{B}
ceil $, where $ \lceil x
ceil $ is the ceiling function that gives the smallest integer greater than or equal to $ x $).
# Answer:
Atsidi can first determine the amount of yarn required to make one triangle (let's call this $ y $). Then, using the formula for the total number of triangles (from part 4b) for a given number of borders ($ n $), she can calculate the total number of triangles ($ T $). The total yarn needed is $ T\times y $. If each yarn bundle contains a known amount of yarn (say $ b $ units), she can divide the total yarn needed by the amount of yarn per bundle ($ \frac{T\times y}{b} $) and round up to the nearest whole number (using the ceiling function) to determine the number of yarn bundles to prepare. For example, if she wants to make $ n = 20 $ borders, she can use the total number of triangles (690 from part 4a), multiply by $ y $ to get total yarn, and then divide by $ b $ (yarn per bundle) to find the number of bundles.

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QUESTION IMAGE

Question

Explanation:

Part 3a:

Step1: Identify the sequence type

Step2: Substitute values for \( n = 8 \)

Step1: Explicit formula for sum of geometric sequence

Step2: Recursive formula

Step1: Definition of exponential growth

Step2: Relate to the problem

Answer:

Part 3b: