Sequences
1. Constructing a Brick Staircase A brick staircase has a total of 30 steps. The bottom step requires 100 bricks. Each successive step requires two less bricks than the prior step.
a. How many bricks are required for the top step?
b. How many bricks are required to build the staircase?
2. the Drury Lane Theater has 25 seats in the first row and 37 in the 7th row. If the number of seats in each row is an arithmetic sequence, how many seats are in the 21th row? (Use the formula to find d and then again to find the number of seats in the 21th row.)
3. Warren wanted to save money to purchase a new car. he started by saving $1 on the first of January. On the first of February, he saved $3. On the first of March he saved $9. So, on the first day of each month, he wanted to save three times as much as he did on the first day of the previous month. If Warren continues his savings pattern, how much will he need to save on the first day of September?
4. A local charity received $8,500 in charitable contributions during the month of January. Because of a struggling economy, it is projected that contributions will decline each month to 95% of the previous month's contributions.
a. What are the expected contributions for the month of August?
b. What is the total expected contributions that this charity can expect at the end of the year?
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what are the answers to number 3
1. Constructing a Brick Staircase:
a. To find how many bricks are required for the top step, we can use the given information that the bottom step requires 100 bricks and each successive step requires two less bricks than the previous step.
We can set up a sequence to represent the number of bricks required for each step. The first term of the sequence is 100, and each term after that is two less than the previous term.
The sequence can be written as: 100, ?, ?, ?, ..., ?
To find the number of bricks required for the top step, we need to find the nth term of this sequence, where n is the total number of steps.
To find the nth term of an arithmetic sequence, we can use the formula:
nth term = first term + (n - 1) * common difference
In this case, the first term is 100 and the common difference is -2 (since each successive step requires two less bricks).
Since we have a total of 30 steps, we can substitute these values into the formula:
nth term = 100 + (30 - 1) * (-2)
nth term = 100 + 29 * (-2)
nth term = 100 - 58
nth term = 42
Therefore, the top step of the brick staircase requires 42 bricks.
b. To find the total number of bricks required to build the staircase, we need to find the sum of the sequence. We know that the sequence has 30 terms, with the first term being 100 and a common difference of -2.
To find the sum of an arithmetic sequence, we can use the formula:
sum = (n/2) * (first term + last term)
In this case, the first term is 100, the last term is 42, and n is 30 (the total number of terms).
Substituting these values into the formula, we get:
sum = (30/2) * (100 + 42)
sum = 15 * 142
sum = 2130
Therefore, the total number of bricks required to build the staircase is 2130.
2. Drury Lane Theater:
To find the number of seats in the 21st row, we are given that the number of seats in each row forms an arithmetic sequence. We need to find the common difference (d) and then use the formula to find the number of seats in the 21st row.
We are given that there are 25 seats in the first row, and 37 seats in the 7th row.
Using the formula for the nth term of an arithmetic sequence:
nth term = first term + (n - 1) * common difference
We can set up two equations using the given information:
25 = first term + (1 - 1) * d
37 = first term + (7 - 1) * d
Simplifying these equations, we get:
25 = first term
37 = first term + 6d
Substituting the value of the first term (25) into the second equation, we have:
37 = 25 + 6d
12 = 6d
d = 2
Therefore, the common difference is 2.
Now, we can use the formula for the nth term to find the number of seats in the 21st row:
nth term = first term + (n - 1) * common difference
Substituting the values of the first term (25), common difference (2), and n (21), we get:
21st row = 25 + (21 - 1) * 2
21st row = 25 + 20 * 2
21st row = 25 + 40
21st row = 65
Therefore, there are 65 seats in the 21st row of the theater.
3. Warren's Savings Pattern:
To find out how much Warren needs to save on the first day of September, we need to determine the pattern of his savings. We are given that on the first of each month, he saves three times as much as he did on the first day of the previous month.
We can set up a sequence to represent the amount Warren saves each month. The first term of the sequence is $1, and each term after that is three times the previous term.
The sequence can be written as: 1, ?, ?, ?, ..., ?
To find the amount Warren needs to save on the first day of September (the ninth month), we need to find the ninth term of this sequence.
To find the nth term of a geometric sequence, we can use the formula:
nth term = first term * (common ratio)^(n - 1)
In this case, the first term is $1, and the common ratio is 3 (since Warren saves three times as much each month).
Since we have nine months, we can substitute these values into the formula:
nth term = 1 * 3^(9 - 1)
nth term = 1 * 3^8
nth term = 1 * 6561
nth term = $6,561
Therefore, Warren needs to save $6,561 on the first day of September.
4. Projected Charitable Contributions:
a. To find the expected contributions for the month of August, we are given that the contributions will decline each month to 95% of the previous month's contributions.
We start with $8,500 in January.
To find the expected contributions for August, we can calculate the contributions for each month starting from January and apply the 95% decline each month.
January: $8,500
February: $8,500 * 0.95 = $8,075
March: $8,075 * 0.95 = $7,671.25
April: $7,671.25 * 0.95 = $7,287.6875
May: $7,287.6875 * 0.95 = $6,923.302
June: $6,923.302 * 0.95 = $6,577.1369
July: $6,577.1369 * 0.95 = $6,248.280
August: $6,248.28 * 0.95 = $5,935.866
Therefore, the expected contributions for the month of August are $5,935.866.
b. To find the total expected contributions that the charity can expect at the end of the year, we need to sum up the contributions for each month.
We already have the values for the contributions from January to August. To find the contributions for the remaining months, we can continue the pattern of multiplying by 0.95.
September: $5,935.866 * 0.95 = $5,638.0727
October: $5,638.0727 * 0.95 = $5,356.1691
November: $5,356.1691 * 0.95 = $5,088.3606
December: $5,088.3606 * 0.95 = $4,834.9426
Now, we can sum up all the contributions:
Total expected contributions = (January + February + ... + December)
Total expected contributions = $8,500 + $8,075 + $7,671.25 + $7,287.6875 + $7,923.302 + $6,577.1369 + $6,248.28 + $5,935.866 + $5,638.0727 + $5,356.1691 + $5,088.3606 + $4,834.9426
Total expected contributions = $84,737.0534
Therefore, the total expected contributions that the charity can expect at the end of the year is approximately $84,737.05.