Robin bought a computer for $1,250. It will depreciate, or decrease in value, by 10% 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

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The numbers of seats in the first 12 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 12th row?

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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 + 10 + 17 + 24 + ...

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 sum

b. Find the total number of beads in the necklace. Explain your method for finding the total number of beads.

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Beads: Add 7 to each row.

3+10+17+24+31+38+45+52+59+66+73+80+87+94+101+108+115+122

= 1,125

bruh. ya'll tweaked out or sum i just want the answer to this summer school sucks.

a. The sequence formed by the value of the computer each year is geometric. Geometric sequences have a common ratio between each term, and in this case, the value of the computer decreases by 10% each year.

b. The explicit formula to represent the sequence is V = 1250 * (0.9)^n, where V 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 can plug in n = 6 into the explicit formula:
V = 1250 * (0.9)^6
V = 1250 * 0.531441
V ≈ 664.30

Therefore, the value of the computer at the beginning of the 6th year is approximately $664.30.

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a. The recursive formula to represent the sequence is: a(n) = a(n-1) + 2, where a(n) represents the number of seats in the nth row.

b. The explicit formula to represent the sequence is: a(n) = 7 + 2n, where a(n) represents the number of seats in the nth row.

c. To find the number of seats in the 12th row, we can plug in n = 12 into the explicit formula:
a(12) = 7 + 2(12)
a(12) = 7 + 24
a(12) = 31

Therefore, there are 31 seats in the 12th row.

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a. In summation notation, the series can be represented as: ∑(n=1 to 18) (7n - 4).

i. The lower limit of the sum is 1, as the series starts from the first row.
ii. The upper limit of the sum is 18, as there are 18 rows.
iii. The explicit formula of the sum is ∑(n=1 to 18) (7n - 4).

b. To find the total number of beads in the necklace, we can evaluate the sum of the series:
∑(n=1 to 18) (7n - 4)
= (7(1) - 4) + (7(2) - 4) + (7(3) - 4) + ... + (7(18) - 4)
= 3 + 10 + 17 + ... + 122
= 1125

Therefore, the total number of beads in the necklace is 1125. I calculated it by evaluating the sum of the series using the formula.

a. The sequence formed by the value of the computer at the beginning of each year is geometric. This is because the value decreases by a constant rate of 10% each year.

b. The explicit formula to represent the sequence can be written as:

Vn = V0 * (1 - r)^n

where Vn represents the value at the beginning of the nth year, V0 represents the initial value of the computer, r represents the rate of depreciation (0.1 for a 10% decrease), and n represents the number of years.

c. To find the value of the computer at the beginning of the 6th year, we substitute n = 6 into the formula from part b:

V6 = 1250 * (1 - 0.1)^6
V6 = 1250 * 0.9^6
V6 ≈ 683.52

Therefore, the value of the computer at the beginning of the 6th year is approximately $683.52.

To solve each of these questions, I will guide you step by step on how to approach them.

a. To determine if the sequence formed by the value of the computer at the beginning of each year is arithmetic, geometric, or neither, we need to look for a pattern.

In this case, the value of the computer decreases by 10% each year. This means each year's value is the previous year's value multiplied by 0.9 (100% - 10%). Therefore, the sequence formed is geometric.

b. To write an explicit formula to represent the sequence, we can use the formula for a geometric sequence. The explicit formula is given by:

An = A1 * r^(n-1)

where An represents the value at the beginning of the nth year, A1 is the initial value of the computer, r is the common ratio (0.9 in this case), and n is the year number.

Therefore, the explicit formula for this sequence is:

An = $1,250 * 0.9^(n-1)

c. To find the value of the computer at the beginning of the 6th year, we substitute n = 6 into the formula we obtained in part b:

A6 = $1,250 * 0.9^(6-1)
= $1,250 * 0.9^5
≈ $915.83

Therefore, the value of the computer at the beginning of the 6th year is approximately $915.83.

Now let's move on to the next question.

a. To write a recursive formula to represent the sequence of the number of seats in each row, we need to look for a pattern. In this case, each row has 2 more seats than the previous row. This means the recursive formula will involve adding 2 to the previous term.

Let's denote the number of seats in the first row as a1. Then we have:

a2 = a1 + 2 (2 more seats than the previous row)
a3 = a2 + 2
a4 = a3 + 2
...
an = an-1 + 2

Therefore, the recursive formula for this sequence is:

an = an-1 + 2

b. To write an explicit formula to represent the sequence, we can use the formula for an arithmetic sequence. The explicit formula is given by:

an = a1 + (n-1)d

where an represents the number of seats in the nth row, a1 is the number of seats in the first row, n is the row number, and d is the common difference (2 in this case).

Therefore, the explicit formula for this sequence is:

an = 9 + (n-1)2
= 9 + 2n - 2
= 7 + 2n

c. To find the number of seats in the 12th row, we substitute n = 12 into the formula we obtained in part b:

a12 = 7 + 2(12)
= 7 + 24
= 31

Therefore, there are 31 seats in the 12th row.

Lastly, let's move on to the third question.

a. The series of the number of beads per row can be represented using summation notation. Summation notation is a concise way to represent the sum of a sequence.

i. The lower limit of the sum is the first term of the series, which is 3.
ii. The upper limit of the sum is the last term of the series. To figure this out, we need to find a pattern. In this case, we can observe that each term is 7 more than the previous term. Therefore, we can find the last term by finding the nth term that is greater than the number of rows (18).

To do this, we solve the equation 3 + 7(n-1) > 18, where n represents the number of terms in the series. Solving this inequality, we find:

3 + 7(n-1) > 18
7(n-1) > 15
n-1 > 15/7
n > 2.14

Since n must be a whole number, n = 3 will be the smallest value greater than 2.14 that satisfies the inequality. Therefore, the upper limit of the sum is 3.

iii. The explicit formula of the sum can be written as:

∑ (3 + 7(n-1)), for n = 1 to 3

b. To find the total number of beads in the necklace, we can calculate the sum represented by the summation in part a.

∑ (3 + 7(n-1)), for n = 1 to 3
= (3 + 7(1-1)) + (3 + 7(2-1)) + (3 + 7(3-1))
= 3 + 3 + 10 + 3 + 17
= 36

Therefore, the total number of beads in the necklace is 36.

computer:

(a) geometric, since each term is 0.90 times the last
(b) 1250(0.90)^(n-1)
(c) 1250(0.90)^5

seats:
a=9
d = 11-9 = 2
a_n = a_(n-1) + 2
a_n = 9 + (n-1)*2 = 2n+7

seats:

17
∑ 3+7k
k=0

S18 = 18/2 (3 + 17*7)