5. The local soccer team won the championship a several years ago, and since then, ticket prices have been increasing $20 per year. The year they won the championship, tickets were $50. Write a recursive formula for a sequence that will model ticket prices. Is the sequence arithmetic or geometric?
8. A radioactive substance decreases in the amount of grams by one third each year. If the starting amount of the substance in a rock is 1,452 g, write a recursive formula for a sequence that models the amount of the substance left after the end of each year. Is the sequence arithmetic or geometric?
#5:
T1 = 50
Tn+1 = Tn+20
#6:
T1 = 1452
Tn+1 = Tn * 2/3
I expect you can tell an AP from a GP
To write a recursive formula for the sequence that models ticket prices, we need to identify the pattern. We know that ticket prices have been increasing $20 per year since the year the local soccer team won the championship. Let's assume "n" represents the number of years since they won the championship.
The initial ticket price, when n = 0, is $50. In the first year (n = 1), the price increases by $20, making it $70. In the second year (n = 2), it increases by another $20, making it $90. We can observe that each year the ticket price increases by $20.
Therefore, the recursive formula for the ticket prices is:
P(n) = P(n-1) + $20
This formula states that the ticket price in year "n" (P(n)) is equal to the ticket price in the previous year (P(n-1)) plus $20.
As for the sequence being arithmetic or geometric, it is considered arithmetic because each term (ticket price) is obtained by adding a constant amount ($20) to the previous term.
Now, let's move on to the second question about the radioactive substance.
To write a recursive formula for the sequence that models the amount of the radioactive substance left after each year, we need to identify the pattern. We know that the substance decreases by one-third each year.
Let's assume "n" represents the number of years that have passed since the initial amount.
The initial amount of the substance, when n = 0, is 1,452 g. In the first year (n = 1), it decreases by one-third, resulting in 1,452 g * (2/3) = 968 g. In the second year (n = 2), it decreases by another one-third, resulting in 968 g * (2/3) = 645.33 g (rounded to two decimal places).
Therefore, the recursive formula for the amount of the substance left after each year is:
A(n) = A(n-1) * (2/3)
This formula states that the amount of the substance left in year "n" (A(n)) is equal to the amount of the substance left in the previous year (A(n-1)) multiplied by two-thirds.
As for the sequence being arithmetic or geometric, it is considered geometric because each term (amount of substance left) is obtained by multiplying the previous term by a constant ratio (2/3).