High School

We appreciate your visit to Solve each problem using your knowledge of sequences You may encounter both arithmetic and geometric sequences 1 You visit the Grand Canyon and drop a. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

Solve each problem using your knowledge of sequences. You may encounter both arithmetic and geometric sequences.

1. You visit the Grand Canyon and drop a penny off the edge of a cliff. The distance the penny will fall is 16 feet in the first second, 48 feet in the second second, 80 feet in the third second, and so on, in an arithmetic sequence. What is the total distance the object will fall in 6 seconds?

Please type your answer here.

2. After knee surgery, your trainer advises you to return to jogging gradually. He suggests jogging for 12 minutes each day for the first week. Each subsequent week, he recommends increasing that time by 6 minutes per day. How many weeks will it take before you are jogging 60 minutes per day?

Please type your answer here.

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.

Thanks for taking the time to read Solve each problem using your knowledge of sequences You may encounter both arithmetic and geometric sequences 1 You visit the Grand Canyon and drop a. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada