Answer :
Sure, let's break down the solutions for both parts of the question step-by-step:
### Problem 1: Total Distance the Penny Will Fall
You have an arithmetic sequence where the distance the penny falls each second forms the sequence: 16 feet, 48 feet, 80 feet, etc. This is an arithmetic sequence because the difference between each term is constant.
1. Identify the first term and common difference:
- First term ([tex]\(a_1\)[/tex]) = 16 feet
- Common difference ([tex]\(d\)[/tex]) = 48 feet - 16 feet = 32 feet
2. Number of terms ([tex]\(n\)[/tex]):
- We need to find the total distance for 6 seconds.
3. Sum of an arithmetic sequence formula:
- The formula for the sum of the first [tex]\(n\)[/tex] terms of an arithmetic sequence is:
[tex]\[
S_n = \frac{n}{2} \times (2a_1 + (n-1)d)
\][/tex]
4. Plug in the values:
- [tex]\(n = 6\)[/tex]
- [tex]\(a_1 = 16\)[/tex]
- [tex]\(d = 32\)[/tex]
[tex]\[
S_6 = \frac{6}{2} \times (2 \times 16 + (6-1) \times 32) = 576 \text{ feet}
\][/tex]
So, the total distance the penny will fall in 6 seconds is 576 feet.
### Problem 2: Weeks to Jog 60 Minutes per Day
The sequence for the jogging program is also arithmetic, starting at 12 minutes and increasing by 6 minutes each week.
1. Identify the known terms:
- First term ([tex]\(a_1\)[/tex]) = 12 minutes
- Common difference ([tex]\(d\)[/tex]) = 6 minutes
- Final desired term ([tex]\(a_n\)[/tex]) = 60 minutes
2. Arithmetic sequence formula to find the term position ([tex]\(n\)[/tex]):
- The [tex]\(n\)[/tex]-th term of an arithmetic sequence is given by:
[tex]\[
a_n = a_1 + (n-1) \times d
\][/tex]
3. Solve for [tex]\(n\)[/tex]:
- Plug in the known values:
[tex]\[
60 = 12 + (n-1) \times 6
\][/tex]
- Simplify and solve for [tex]\(n\)[/tex]:
[tex]\[
60 = 12 + 6n - 6 \\
60 = 6 + 6n \\
54 = 6n \\
n = \frac{54}{6} = 9
\][/tex]
It will take 9 weeks to reach jogging 60 minutes per day.
I hope this explanation clarifies things! If you have more questions, feel free to ask.
### Problem 1: Total Distance the Penny Will Fall
You have an arithmetic sequence where the distance the penny falls each second forms the sequence: 16 feet, 48 feet, 80 feet, etc. This is an arithmetic sequence because the difference between each term is constant.
1. Identify the first term and common difference:
- First term ([tex]\(a_1\)[/tex]) = 16 feet
- Common difference ([tex]\(d\)[/tex]) = 48 feet - 16 feet = 32 feet
2. Number of terms ([tex]\(n\)[/tex]):
- We need to find the total distance for 6 seconds.
3. Sum of an arithmetic sequence formula:
- The formula for the sum of the first [tex]\(n\)[/tex] terms of an arithmetic sequence is:
[tex]\[
S_n = \frac{n}{2} \times (2a_1 + (n-1)d)
\][/tex]
4. Plug in the values:
- [tex]\(n = 6\)[/tex]
- [tex]\(a_1 = 16\)[/tex]
- [tex]\(d = 32\)[/tex]
[tex]\[
S_6 = \frac{6}{2} \times (2 \times 16 + (6-1) \times 32) = 576 \text{ feet}
\][/tex]
So, the total distance the penny will fall in 6 seconds is 576 feet.
### Problem 2: Weeks to Jog 60 Minutes per Day
The sequence for the jogging program is also arithmetic, starting at 12 minutes and increasing by 6 minutes each week.
1. Identify the known terms:
- First term ([tex]\(a_1\)[/tex]) = 12 minutes
- Common difference ([tex]\(d\)[/tex]) = 6 minutes
- Final desired term ([tex]\(a_n\)[/tex]) = 60 minutes
2. Arithmetic sequence formula to find the term position ([tex]\(n\)[/tex]):
- The [tex]\(n\)[/tex]-th term of an arithmetic sequence is given by:
[tex]\[
a_n = a_1 + (n-1) \times d
\][/tex]
3. Solve for [tex]\(n\)[/tex]:
- Plug in the known values:
[tex]\[
60 = 12 + (n-1) \times 6
\][/tex]
- Simplify and solve for [tex]\(n\)[/tex]:
[tex]\[
60 = 12 + 6n - 6 \\
60 = 6 + 6n \\
54 = 6n \\
n = \frac{54}{6} = 9
\][/tex]
It will take 9 weeks to reach jogging 60 minutes per day.
I hope this explanation clarifies things! If you have more questions, feel free to ask.
