80. Determine the sum for the arithmetic series:
n = 37, t1 = 23, t37 = 203
a. 5631
b. 8362
c. 6742
d. 4181
81. Determine the sum for the arithmetic series:
n = 100, t1 = 17, t100 = 215
a. 9900
b. 3350
c. 11,600
d. 990
82. Determine the sum for the geometric series:
n=10, r=-2, t1=1
a. -341
b. 9
c. 228
d. -400
83.Find S20 if the series 1+1.1+... is arithmetic.
a. 2.9
b. 1.9
c. 39
d. 78
84. Determine the sum for the positive two-digit integers that are not divisible by 5.
a. 3960
b. 1640
c. 2830
d. 4550
85. A ship's clock strikes every half hour of a 4-hour period. After the first half hour it strikes "one bell," after another half hour it strikes "two bells," and so on until it strikes "eight bells" at the end of the four hour period. It then begins a new four hour period . How many strikes of the bell occur in one day?
a. 36
b. 216
c. 192
d. 84
87. Sketch the angle in standard position, indicating its rotation by a curved arrow. Choose the quadrant where the angle is located. Angle: 240°
a. III
b. I
c. IV
d. II
88. Determine the correct formula for the measures of all angles coterminal with the angle: 140°
a. 140° + (360° + n)
b. 140° + n(360°)
c. 360° + n(180°)
d. 360° + (180° + n)
80. The sum for the arithmetic series is given by the formula: Sn = (n/2)(t1 + tn), where Sn is the sum, n is the number of terms, t1 is the first term, and tn is the last term. Plugging in the values, we get Sn = (37/2)(23 + 203) = 6742. Therefore, the answer is c. 6742.
81. Using the same formula, we get Sn = (100/2)(17 + 215) = 9900. So, the answer is a. 9900.
82. The sum for a geometric series is given by the formula: Sn = (t1(1 - r^n))/(1 - r), where Sn is the sum, t1 is the first term, r is the common ratio, and n is the number of terms. Plugging in the values, we get Sn = (1(1 - (-2)^10))/(1 - (-2)) = 1023/3 = 341. Therefore, the answer is a. -341.
83. Since the series 1+1.1+... is arithmetic, we can use the formula for the sum of an arithmetic series: Sn = (n/2)(t1 + tn), where Sn is the sum, n is the number of terms, t1 is the first term, and tn is the last term. Plugging in the values, we get Sn = (20/2)(1 + 1.1) = 39. Therefore, the answer is c. 39.
84. To find the sum of positive two-digit integers that are not divisible by 5, we need to sum the numbers from 10 to 99 that are not divisible by 5. We can do this by finding the sum of all two-digit integers from 10 to 99 and subtracting the sum of the two-digit integers that are divisible by 5. The sum of all two-digit integers from 10 to 99 is (10 + 99)*(90/2) = 4950. The sum of the two-digit integers that are divisible by 5 is (15 + 95)*(81/2) = 3780. Therefore, the sum of positive two-digit integers that are not divisible by 5 is 4950 - 3780 = 1170. However, none of the options provided match this answer, so none of the options are correct.
85. In a four-hour period, the ship's clock strikes "one bell" after the first half hour, "two bells" after the second half hour, and so on until it strikes "eight bells" at the end. This means that the ship's clock strikes a total of 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 times in a four-hour period. In one day, there are 24 hours, so the ship's clock will strike 36 * (24/4) = 216 times. Therefore, the answer is b. 216.
87. To sketch the angle in standard position, we start by drawing the positive x-axis (the horizontal axis) and the positive y-axis (the vertical axis). Then, we rotate 240° counterclockwise from the positive x-axis. This angle is located in the third quadrant. So, the answer is a. III.
88. An angle is coterminal with another if they have the same initial side and terminal side. To find angles coterminal with 140°, we can add or subtract multiples of 360°. Therefore, the correct formula is b. 140° + n(360°).
80. To determine the sum for the arithmetic series, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(t1 + tn)
where Sn is the sum of the series, n is the number of terms, t1 is the first term, and tn is the last term.
Substituting the provided values:
n = 37, t1 = 23, tn = 203
Sn = (37/2)(23 + 203)
Sn = (37/2)(226)
Sn = 37(113)
Sn = 4181
Therefore, the sum for the arithmetic series is 4181.
Answer: d. 4181
81. Similar to the previous question, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(t1 + tn)
Substituting the provided values:
n = 100, t1 = 17, tn = 215
Sn = (100/2)(17 + 215)
Sn = (100/2)(232)
Sn = 50(232)
Sn = 11600
Therefore, the sum for the arithmetic series is 11600.
Answer: c. 11600
82. To determine the sum for the geometric series, we can use the formula:
Sn = (t1 * (r^n - 1))/(r - 1)
where Sn is the sum of the series, t1 is the first term, r is the common ratio, and n is the number of terms.
Substituting the provided values:
n = 10, t1 = 1, r = -2
Sn = (1 * (-2^10 - 1))/(-2 - 1)
Sn = (-1023)/(-3)
Sn = 341
Therefore, the sum for the geometric series is 341.
Answer: a. -341
83. If the given series is arithmetic, we can calculate the common difference (d) and then use the formula for the sum of an arithmetic series.
For an arithmetic series, the common difference is given by the formula:
d = t2 - t1
Given that the series 1 + 1.1 + ... is arithmetic:
d = (1.1 - 1) = 0.1
With the common difference, we can calculate the sum of the first 20 terms using the formula:
Sn = (n/2)(t1 + tn)
Substituting the values:
n = 20, t1 = 1
Sn = (20/2)(1 + t20)
Sn = 10(1 + 1.1)
Sn = 10(2.1)
Sn = 21
Therefore, S20 (the sum of the first 20 terms) is 21.
Answer: N/A (This question doesn't provide valid answer choices)
84. To determine the sum for the positive two-digit integers that are not divisible by 5, we need to find the numbers that meet this criteria and then calculate their sum.
The positive two-digit integers that are not divisible by 5 are: 11, 12, 13, 16, 17, 18, 19, 21, 22, 23, 26, 27, 28, 29, 31, 32, 33, 36, 37, 38, 39, 41, 42, 43, 46, 47, 48, 49, 51, 52, 53, 56, 57, 58, 59, 61, 62, 63, 66, 67, 68, 69, 71, 72, 73, 76, 77, 78, 79, 81, 82, 83, 86, 87, 88, 89, 91, 92, 93, 96, 97, 98, 99
To find the sum, we can use the formula for the sum of an arithmetic series:
Sn = (n/2)(t1 + tn)
where Sn is the sum of the series, n is the number of terms, t1 is the first term, and tn is the last term.
Substituting the values:
n = 54 (number of terms), t1 = 11, tn = 99
Sn = (54/2)(11 + 99)
Sn = (54/2)(110)
Sn = 27(110)
Sn = 2970
Therefore, the sum for the positive two-digit integers that are not divisible by 5 is 2970.
Answer: N/A (This question doesn't provide valid answer choices)
85. To find the total number of strikes of the bell in one day, we can calculate the strikes that occur within one four-hour period and then multiply it by the number of four-hour periods in a day.
In one 4-hour period:
- After the first half hour, it strikes "one bell."
- After another half hour, it strikes "two bells."
- And so on, until it strikes "eight bells" at the end of the four-hour period.
Therefore, the total number of strikes in one 4-hour period is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 strikes.
In one day, there are 24 hours. Since each four-hour period consists of 36 strikes, the number of four-hour periods in one day is 24/4 = 6.
Therefore, the total number of strikes in one day is 36 strikes * 6 periods = 216 strikes.
Answer: b. 216
87. To sketch the angle in standard position, we start by placing the initial side of the angle on the positive x-axis and rotate counterclockwise.
Angle: 240°
To determine the quadrant where the angle is located, we can use the following guidelines:
- If the angle is between 0° and 90°, it is in Quadrant I.
- If the angle is between 90° and 180°, it is in Quadrant II.
- If the angle is between 180° and 270°, it is in Quadrant III.
- If the angle is between 270° and 360°, it is in Quadrant IV.
In this case, since the angle is 240°, it falls in Quadrant III.
Answer: c. IV
88. To find the formula for coterminal angles with 140°, we can add or subtract multiples of 360° from 140°.
The correct formula for coterminal angles with 140° is given by:
140° + (360° * n)
where n can be any integer.
Answer: a. 140° + (360° + n)
To solve the problems, let's start by understanding the concepts and formulas involved.
Arithmetic Series:
The sum of an arithmetic series can be found using the formula: Sn = (n/2)(t1 + tn), where Sn represents the sum of the series, n is the number of terms, t1 is the first term, and tn is the nth term.
Geometric Series:
The sum of a geometric series can be found using the formula: Sn = (t1 * (r^n - 1)) / (r - 1), where Sn represents the sum of the series, n is the number of terms, t1 is the first term, and r is the common ratio.
Coterminal Angles:
Coterminal angles are angles that have the same initial and terminal sides but differ by a multiple of 360 degrees. So, to find angles coterminal with a given angle, you can add or subtract multiples of 360 degrees.
Now, let's solve the problems one by one:
80. For the arithmetic series with n = 37, t1 = 23, and t37 = 203:
Using the formula, Sn = (n/2)(t1 + tn), we can substitute the values: Sn = (37/2)(23 + 203).
Calculating, Sn = 18(226) = 4068.
Therefore, the sum of the arithmetic series is 4068.
81. For the arithmetic series with n = 100, t1 = 17, and t100 = 215:
Using the formula, Sn = (n/2)(t1 + tn), we can substitute the values: Sn = (100/2)(17 + 215).
Calculating, Sn = 50(232) = 11,600.
Therefore, the sum of the arithmetic series is 11,600.
82. For the geometric series with n = 10, r = -2, and t1 = 1:
Using the formula, Sn = (t1 * (r^n - 1)) / (r - 1), we can substitute the values: Sn = (1 * (-2^10 - 1)) / (-2 - 1).
Calculating, Sn = (-1023) / (-3) = 341.
Therefore, the sum of the geometric series is 341.
83. If the series 1+1.1+... is arithmetic, we can find the common difference (d) by subtracting the second term from the first term: 1.1 - 1 = 0.1.
Since the common difference remains the same throughout, we can use the formula for arithmetic series.
Using the formula, Sn = (n/2)(t1 + tn), we can substitute the values: Sn = (20/2)(1 + (1 + (20 - 1) * 0.1)).
Calculating, Sn = 10(3) = 30.
Therefore, S20 (the sum of the series with 20 terms) is 30.
84. To find the sum of positive two-digit integers not divisible by 5:
First, we need to identify the range of two-digit integers that are not divisible by 5. This range is from 10 to 99, inclusive.
We can use the formula for the sum of an arithmetic series to find the sum.
The first term, t1, is the smallest two-digit integer not divisible by 5, which is 11.
The last term, tn, is the largest two-digit integer not divisible by 5, which is 99.
Using the formula, Sn = (n/2)(t1 + tn), we can substitute the values: Sn = (90/2)(11 + 99).
Calculating, Sn = 45(110) = 4950.
Therefore, the sum of positive two-digit integers not divisible by 5 is 4950.
85. In a 4-hour period, the ship's clock strikes every half hour. To find the total number of strikes in one day (24 hours):
Since the clock strikes every half hour and there are 24 hours in a day, there will be a total of 24*2 = 48 half hours in a day.
The pattern repeats every 4 hours, so we need to find the number of strikes in a single 4-hour period and multiply it by 6 (24/4).
In the first half hour, it strikes "one bell." In the second half hour, it strikes "two bells." This pattern continues until the last half hour, where it strikes "eight bells."
The sum of the sequence 1+2+3+...+8 can be found using the formula Sn = (n/2)(t1 + tn), where n = 8, t1 = 1, and tn = 8.
Using the formula, Sn = (8/2)(1 + 8) = 4(9) = 36.
Therefore, within a 4-hour period, the ship's clock strikes 36 times.
Multiplying by 6 gives the total number of strikes in one day: 36 * 6 = 216.
Therefore, the number of strikes of the bell in one day is 216.
87. To plot the angle 240° in standard position and determine the quadrant:
Start by drawing the x and y axes. The initial side of the angle will coincide with the positive x-axis.
Next, rotate 240° counter-clockwise from the initial side, represented by a curved arrow in the clockwise direction.
The angle of 240° falls in the third quadrant.
Therefore, the angle 240° is located in Quadrant III.
88. To find the formula for angles coterminal with 140°, we add or subtract multiples of 360.
The formula for coterminal angles is given by 140° + n(360°), where n represents any integer.
Therefore, the correct formula for coterminal angles with 140° is option b: 140° + n(360°).