I need help on these questions:
1. After returning from a knee injury, your trainer tells you to return to your running program slowly. She suggests running for 60 minutes total for the first week. Each week thereafter, she suggests that you increase that time by 6 minutes each week. What type of sequence best models this scenario?
-Arithmetic Sequences
-Geometric Sequence
-Neither
2. Which sequence best models the scenario from Question 1?
-60, 54, 48, 42, ...
-60, 66, 72, 78, ...
-60, 360, 2160, ...
-54, 60, 66, 72, ...
3. Write a recursive equation to model the sequence from question 2
-t(n + 1) = t(n) + 6, t(1) = 0
-t(n) = 6n + 54
-t(n + 1) = t(n) + 6
-t(n + 1) = t(n) + 6, t(1) = 60
4. A professional basketball player signs a contract with a beginning salary of $3,000,000 for the first year and an annual increase of 4% per year. That is, beginning in year 2, the athlete's salary will be 1.04 times what it was in the previous year. What type of sequence best models the scenario?
-Geometric Sequence
-Neither
-Arithmetic Sequence
-No answer text provided.
5. Which sequence best models the scenario from question 4.
-3,000,000 , 120,000 , 4,800 , 192 , ...
-3,000,000 , 3,004,000 , 3,008,000, ...
-3,000,000 , 3,000,004 , 3,000,008, ...
-3,000,000 , 3,120,000 , 3,444,800, ...
6. Write a recursive equation to model the scenario and sequence from questions 4 and 5.
-t(n + 1) = 1.04•t(n)
-t(n) = 3,000,000(1.04)^n
-t(n + 1) = 1.04•t(n) , t(1) = 3,000,000
-t(n + 1) = t(n) + 1.04 , t(1) = 3,000,000
https://www.jiskha.com/questions/1821390/1-after-returning-from-a-knee-injury-your-trainer-tells-you-to-return-to-your-running
Can somebody check my math before I submit it? Thanks.
1. Answer: Arithmetic Sequences
2. Answer: 60, 66, 72, 78, ...
3. Answer: t(n + 1) = t(n) + 6, t(1) = 60
4. Answer: Geometric Sequence
5. Answer: 3,000,000 , 3,120,000 , 3,444,800, ...
6. Answer: t(n + 1) = t(n) + 1.04 , t(1) = 3,000,000
all but 6
t(n) = 3,000,000(1.04)^n
6 is also correct, since they asked for a recursion formula.
1. To determine the type of sequence that best models the scenario, we need to identify whether there is a fixed common difference or a fixed common ratio. In arithmetic sequences, there is a fixed common difference between consecutive terms, while geometric sequences have a fixed common ratio.
In this scenario, the trainer suggests increasing the running time by 6 minutes each week. Since there is a fixed increase of 6 minutes, the scenario can be best modeled by an Arithmetic Sequence.
2. Now we need to identify which of the given sequences best models the scenario from Question 1. Let's examine each option:
- Sequence 1: 60, 54, 48, 42,...
- Sequence 2: 60, 66, 72, 78,...
- Sequence 3: 60, 360, 2160,...
- Sequence 4: 54, 60, 66, 72,...
Looking at the scenarios, it is clear that the only sequence that follows the given pattern of starting with 60 and each subsequent term decreasing by 6 is Sequence 1: 60, 54, 48, 42,.... Therefore, Sequence 1 best models the scenario from Question 1.
3. To write a recursive equation to model the sequence from Question 2, we need to determine the pattern between consecutive terms.
In this case, the pattern is that each term is obtained by taking the previous term and adding 6 to it. So the recursive equation would be:
t(n + 1) = t(n) + 6, with the initial term t(1) = 60.
Therefore, the correct answer is:
- t(n + 1) = t(n) + 6, t(1) = 60.
4. To determine the type of sequence that best models the scenario described in Question 4, we need to identify whether there is a fixed common difference or a fixed common ratio. In arithmetic sequences, there is a fixed common difference between consecutive terms, while geometric sequences have a fixed common ratio.
In this scenario, the athlete's salary increases by 4% each year. Since there is a fixed multiplication factor of 1.04, the scenario can be best modeled by a Geometric Sequence.
5. Now we need to identify which of the given sequences best models the scenario from Question 4. Let's examine each option:
- Sequence 1: 3,000,000, 120,000, 4,800, 192,...
- Sequence 2: 3,000,000, 3,004,000, 3,008,000,...
- Sequence 3: 3,000,000, 3,000,004, 3,000,008,...
- Sequence 4: 3,000,000, 3,120,000, 3,444,800,...
Looking at the scenarios, it is clear that the only sequence that follows the given pattern of starting with 3,000,000 and each subsequent term being 1.04 times the previous term is Sequence 2: 3,000,000, 3,004,000, 3,008,000, .... Therefore, Sequence 2 best models the scenario from Question 4.
6. To write a recursive equation to model the scenario and sequence from Questions 4 and 5, we need to determine the pattern between consecutive terms.
In this case, each term is obtained by taking the previous term and multiplying it by 1.04. So the recursive equation would be:
t(n + 1) = 1.04•t(n), with the initial term t(1) = 3,000,000.
Therefore, the correct answer is:
- t(n + 1) = 1.04•t(n), t(1) = 3,000,000.