1. Robin bought a computer for $1,250. It will depreciate, or decrease in value, by 20% each year that she owns it.
a. Is the sequence formed by the value at the beginning of each year arithmetic, geometric, or neither? Explain.
b. Write an explicit formula to represent the sequence.
c. Find the value of the computer at the beginning of the 6th year.
2. The numbers of seats in the first 10 rows of a high-school auditorium form an arithmetic sequence. The first row has 9
seats. The second row has 11 seats.
a. Write a recursive formula to represent the sequence.
b. Write an explicit formula to represent the sequence.
c. How many seats are in the 10th row?
3. Dante is making a necklace with 18 rows of tiny beads in which the number of beads per row is given by the series 3
+ 9 + 15 + 21 + ...
a. If you were to write this series in summation notation, give
i. the lower limit of the sum
ii. the upper limit of the sum
iii. the explicit formula of the series
b. Find the total number of beads in the necklace. Explain your method for finding the total number of beads.
1.
a. The sequence formed by the value of the computer each year is geometric because it is decreasing by a constant percentage (20%) each year.
b. The explicit formula to represent the sequence is: Vn = 1250(0.8)^n, where Vn is the value of the computer at the beginning of the nth year.
c. To find the value of the computer at the beginning of the 6th year, we plug n = 6 into the formula: V6 = 1250(0.8)^6 = 1250(0.26214) ≈ $327.68
2.
a. The recursive formula to represent the sequence is: a(n) = a(n-1) + 2, where a(1) = 9.
b. The explicit formula to represent the sequence is: a(n) = 7 + 2(n-1).
c. To find the number of seats in the 10th row, we plug n = 10 into the explicit formula: a(10) = 7 + 2(10-1) = 7 + 18 = 25 seats.
3.
a.
i. The lower limit of the sum is 1, as it starts from the first row of beads.
ii. The upper limit of the sum is 18, as there are 18 rows of beads.
iii. The explicit formula of the series is: a(n) = 3 + 6(n-1), where a(1) = 3.
b. To find the total number of beads in the necklace, we need to sum up the series from n = 1 to n = 18: Σ(3 + 6(n-1)), where n goes from 1 to 18. This will give us the total number of beads in the necklace.