1. Robin bought a computer for $1,250. It will depreciate, or decrease in value, by 20% each year that she owns it.

Question

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.

Answer 1

1.
a. The sequence is geometric because the value of the computer decreases by a constant percentage each year.
b. The explicit formula for the value of the computer at the beginning of each year is $1250 * (0.8)^n, where n is the number of years.
c. To find the value of the computer at the beginning of the 6th year, we plug in n=6 into the formula: $1250 * (0.8)^6 = $483.84.

2.
a. The recursive formula for the number of seats in each row is:
a_n = a_{n-1} + 2, where a_1 = 9.
b. The explicit formula for the number of seats in the nth row is: a_n = 7 + 2n.
c. To find the number of seats in the 10th row, we plug in n=10 into the explicit formula: a_10 = 7 + 2(10) = 27 seats.

3.
a.
i. The lower limit of the sum is 1.
ii. The upper limit of the sum is 18.
iii. The explicit formula of the series is: 3 + 6(n-1), where n represents the row number.
b. To find the total number of beads in the necklace, we can use the sum of an arithmetic sequence formula: S_n = (n/2)(a_1 + a_n), where S_n is the total sum, n is the number of terms, a_1 is the first term, and a_n is the last term.
Substitute n=18, a_1=3, and a_n=3 + 6(18-1) into the formula to find the total number of beads.