A student has taken three math tests so far this semester. His scores for the first three tests were 74, 78, and 82. a. Suppose his test scores continue to improve at the same rate. What will be his grade on the sixth (and final) test? b. What will be his total score for all six tests? a. The grade on the final test will be
86. b. His total score for all six tests will be 492.
Explanation:
a. To find the grade on the final test, we can first calculate the difference between each pair of consecutive test scores:
- Difference between test 1 and 2: 78 - 74 = 4
- Difference between test 2 and 3: 82 - 78 = 4
Since the scores are improving at the same rate, we can assume that the difference between the 3rd and 4th test scores will also be 4, and so will the difference between the 4th and 5th test scores. Therefore, the 4th test score will be 86 (82 + 4), and the 5th test score will be 90 (86 + 4). Following the same pattern, the final test score will be 94 (90 + 4).
b. To find the total score for all six tests, we can simply add up all the individual test scores:
74 + 78 + 82 + 86 + 90 + 94 = 492
An amphitheatre has 25 seats in the first row, 27 in the second, 29 in the third, and so on, for 32 rows. How many seats are in the amphitheater?
The amphitheater has seats.
To find the total number of seats in the amphitheater, we need to sum the number of seats in each row. The number of seats in each row increases by 2, so we can write the number of seats in row n as:
25 + (n-1) x 2
where n is the row number.
We want to find the total number of seats in all 32 rows. We can use the formula for the sum of an arithmetic series, which is:
S = (n/2) x (a + l)
where S is the sum, n is the number of terms (in this case, 32), a is the first term (25 seats in the first row), and l is the last term (which we need to find).
We can find l by plugging in n=32 into our formula for the number of seats in a given row:
l = 25 + (32-1) x 2 = 87
Now we can plug in the values for n, a, and l into the formula for the sum:
S = (32/2) x (25 + 87) = 16 x 112 = 1792
Therefore, there are 1792 seats in the amphitheater.
Determine whether the following sequence is geometric. If so, find the common ratio.
1, 5, 25, 125, ...
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
This sequence is geometric with a common ratio of 5.
Explanation:
To determine if a sequence is geometric, we need to check if there is a constant ratio between consecutive terms. We can divide any term by the previous term to check for this ratio:
5/1 = 5
25/5 = 5
125/25 = 5
...
Since the ratio is constant and equal to 5, this sequence is geometric.
Find the fourth term of the geometric sequence 5, 20, 80,
To find the fourth term of a geometric sequence, we need to know the common ratio and one of the terms.
In this sequence, we can find the common ratio by dividing any term by the previous term:
20/5 = 4
80/20 = 4
So the common ratio is 4.
Using the first term, we can find the fourth term by multiplying by the common ratio three times (since we want the fourth term, which is three terms after the first):
5 x 4 x 4 x 4 = 320
Therefore, the fourth term of the sequence is 320.
Find the missing term of the following geometric sequence. It could be the geometric mean or its opposite
32,_,2β¦β¦.
The possible values for the missing term are (Use a comma to separate answers as needed.)
To find the missing term in a geometric sequence, we can check if there is a constant ratio between consecutive terms.
We can divide the second term by the first term to check for the common ratio:
_ / 32 = x so first letβs find x and then find the possible missing values after that.
2 / _ = 1/x
x(_)=32
x = sqrt(32*2)
x = sqrt(64)
x = 8
So the common ratio is 8.
Now we can find the missing term by multiplying the first term by the common ratio and then dividing by the common ratio we have:
32 x 8 / 2 = 128
Therefore, the missing term is 128, and the possible values according to the question are the geometric mean and its opposite, which are:
- The geometric mean of 32 and 2 is sqrt(32 x 2) = sqrt(64) = 8
- The opposite of the geometric mean is -8.
Find the missing term of the following geometric sequence. It could be the geometric mean or its opposite.
12,__,3,β¦
The possible values for the missing term are (Use a comma to separate answers as needed.)
To find the missing term in a geometric sequence, we can check if there is a constant ratio between consecutive terms.
We can divide the second term by the first term to check for the common ratio:
_ / 12 = x so first letβs find x and then find the possible missing values after that.
3 / _ = 1/x
x(_) = 12
x = sqrt(12*3)
x = sqrt(36)
x = 6
So the common ratio is 6.
Now we can find the missing term by multiplying the first term by the common ratio and then dividing by the common ratio we have:
12 x 6 / 3 = 24
Therefore, the missing term is 24, and the possible values according to the question are the geometric mean and its opposite, which are:
- The geometric mean of 12 and 3 is sqrt(12 x 3) = sqrt(36) = 6
- The opposite of the geometric mean is -6.
Find the missing terms of the following geometric sequence. Assume all terms are positive. (Hint: The geometric mean of the first and fifth terms is the third term.) 4,5,__,__,__,364.5,β¦
a2 =
Π°Π· =
a
=
(Type integers or decimals.)
To find the missing terms of a geometric sequence, we need to first determine the common ratio.
We know that the geometric mean of the first and fifth terms is the third term, so we can use this information to find the common ratio:
sqrt(4 x 364.5) = 29.985 (approx.)
Now we can use this common ratio to find the missing terms.
To get from 4 to 5, we multiply by the common ratio:
5 / 4 = 1.24625 (approx.)
So the second term is approximately:
4 x 1.24625 = 4.985 (approx.)
To get from 5 to the third term, we again multiply by the common ratio:
a3 / 5 = 1.24625
a3 = 6.23125 (approx.)
To get from the third term to the fourth term, we multiply by the common ratio again:
a4 / a3 = 1.24625
a4 = 7.7906 (approx.)
To get from the fourth term to the fifth term:
a5 / a4 = 1.24625
a5 = 30.7861 (approx.)
Therefore, the missing terms are approximately:
a2 = 4.985
aΠ· = 6.23125
a4 = 7.7906