1. The explicit form for a sequence is:
an = 4n - 5 .
Write the first four terms in the sequence
2.Write the simplified explicit formula for the arithmetic sequence: -17,-11,-5,1,...
3. The Theater Department is selling tickets to their annual musical production. On the first da of ticket sales they sold 12 adult tickets and 22 student tickets for a total of $92. The schoo took in $40 on the second day by selling 5 adult tickets and 10 student tickets. Write a system of equations to model the situation.
4. Suppose you can work a total of no mare than 15 hours per week at your two jobs Cutting grass pays 10 per hour, and your bagging groceries job pays $7 per hour. You need to AT LEAST $85 per week to cover your expenses. Write a system of inequalities shows the various hours you can work at each job.
#1,2 were already done for you
#3.
12a+22s = 92
5a+10s = 40
#4.
c+b <= 15
10c+7b >= 85
1. To find the first four terms of the sequence given by the explicit form an = 4n - 5, we substitute the values of n in the formula.
For n = 1: a1 = 4(1) - 5 = -1
For n = 2: a2 = 4(2) - 5 = 3
For n = 3: a3 = 4(3) - 5 = 7
For n = 4: a4 = 4(4) - 5 = 11
Therefore, the first four terms of the sequence are -1, 3, 7, 11.
2. To find the explicit formula for the arithmetic sequence -17, -11, -5, 1, ... we need to determine the common difference (d) between consecutive terms.
The common difference (d) can be found by subtracting any two consecutive terms. Here, we can subtract -11 from -17, -5 from -11, and so on:
-17 - (-11) = -17 + 11 = -6
-11 - (-5) = -11 + 5 = -6
-5 - 1 = -6
Since the common difference is -6, we can use this value in the explicit formula:
an = a1 + (n - 1)d
Here, a1 = -17 and d = -6.
Substituting the values, we get:
an = -17 + (n - 1)(-6)
Simplifying this gives the explicit formula for the arithmetic sequence: an = -6n - 11.
3. Let's assume the cost of an adult ticket is x and the cost of a student ticket is y.
On the first day of ticket sales, 12 adult tickets were sold for the cost of 12x, and 22 student tickets were sold for the cost of 22y. The total revenue from the first day is $92, so we have the equation:
12x + 22y = 92
On the second day, 5 adult tickets were sold for the cost of 5x, and 10 student tickets were sold for the cost of 10y. The total revenue from the second day is $40, so we have the equation:
5x + 10y = 40
Therefore, the system of equations to model the situation is:
12x + 22y = 92
5x + 10y = 40
4. Let's assume the number of hours you work cutting grass each week is x, and the number of hours you work bagging groceries is y.
Since you can work a total of no more than 15 hours per week, we have the inequality:
x + y ≤ 15
Cutting grass pays $10 per hour, so the total earned from cutting grass is 10x. Bagging groceries pays $7 per hour, so the total earned from bagging groceries is 7y.
Additionally, you need to earn at least $85 per week to cover your expenses, so we have the inequality:
10x + 7y ≥ 85
Therefore, the system of inequalities that shows the various hours you can work at each job is:
x + y ≤ 15
10x + 7y ≥ 85
1. To find the first four terms in the sequence with the explicit formula an = 4n - 5, you simply need to substitute the values of n from 1 to 4 into the formula:
For n = 1: a1 = 4(1) - 5 = -1
For n = 2: a2 = 4(2) - 5 = 3
For n = 3: a3 = 4(3) - 5 = 7
For n = 4: a4 = 4(4) - 5 = 11
So, the first four terms in the sequence are -1, 3, 7, 11.
2. To find the simplified explicit formula for the given arithmetic sequence -17, -11, -5, 1, ..., you can use the general formula for an arithmetic sequence:
an = a1 + (n - 1)d
where a1 is the first term and d is the common difference.
First, find the common difference between consecutive terms:
-11 - (-17) = 6
-5 - (-11) = 6
1 - (-5) = 6
Since the common difference is always 6, you can write the explicit formula as:
an = -17 + (n - 1)6
So, the simplified explicit formula for the given arithmetic sequence is an = -17 + 6n - 6, which can be further simplified to an = 6n - 23.
3. Let's represent the number of adult tickets on the first day as x, and the number of student tickets as y. We can set up the following system of equations based on the given information:
Equation 1: x + y = 34 (total number of tickets sold on the first day)
Equation 2: 12x + 22y = 92 (total revenue from adult and student tickets on the first day)
Equation 3: 5x + 10y = 40 (total revenue from adult and student tickets on the second day)
So, the system of equations would be:
x + y = 34
12x + 22y = 92
5x + 10y = 40
4. Let's represent the number of hours worked at the grass-cutting job as x, and the number of hours worked at the grocery bagging job as y. We can set up the following system of inequalities to represent the given conditions:
Inequality 1: x + y ≤ 15 (total number of hours worked should not exceed 15 hours per week)
Inequality 2: 10x + 7y ≥ 85 (total earnings from both jobs should be at least $85 per week)
So, the system of inequalities would be:
x + y ≤ 15
10x + 7y ≥ 85
These inequalities represent the various hours you can work at each job while adhering to the given constraints.