Question 1

Solve the problem.
-A brick staircase has a total of 19 steps The first step requires 111 bricks. Each successive step requires 4 fewer bricks than the prior one. How many bricks are required to build the staircase?
A)1425 bricks
B)2850 bricks
C)1387 bricks
D)2793 bricks

Accepted Answer

The number of bricks required forms an arithmetic sequence with the first term $a_1 = 111$ and a common difference $d = -4$. The total number of bricks needed can be found using the formula for the sum of an arithmetic series: $S_n = \frac{n}{2}(2a_1 + (n-1)d)$, where $n = 19$. Substituting the values, we get $S_{19} = \frac{19}{2}(2*111 + (19-1)*(-4)) = \frac{19}{2}(222 - 72) = \frac{19}{2} * 150 = 1425$.

Question 2

Solve the problem.
-A theater has 30 rows with 24 seats in the first row, 28 in the second row, 32 in the third row, and so forth. How many seats are in the theater?
A)4920 seats
B)2460 seats
C)2520 seats
D)5040 seats

Accepted Answer

The answer of Solve the problem.
-A theater has 30 rows...

Question 3

Determine if the sequence is geometric. If the sequence is geometric, find the common ratio.
-3, 12, 48, 192, 768, . . .
A)no
B)yes, 4 
C) yes,  $\frac { 1 } { 4 }$
D)yes, -4

Accepted Answer

The sequence is geometric because each term is obtained by multiplying the previous term by 4.

Question 4

Solve the problem.
-A person puts $20 into a bank account on January 1, $25 on February 1, $30 on March 1, and so forth. How much has the person put into the bank account by December 30?
A)$450
B)$600
C)$1140
D)$570

Accepted Answer

This is an arithmetic sequence where the first term, a = $20, the common difference, d = $5, and the number of terms, n = 12 (for the 12 months). The sum of an arithmetic series is given by S = n/2[2a + (n-1)d]. Substituting the given values, S = 12/2[2(20) + (12-1)5] = 6[40 + 55] = 6 * 95 = $570.

Question 5

Solve the problem.
-As part of her retirement savings plan, Patricia deposited $100 in a bank account during her first year in the workforce. During each subsequent year, she deposited $45 more than the previous year. Find how much she deposited during her twentieth year in the workforce. Find the total amount deposited in the twenty years.
A)$955; $10,550
B)$1000; $22,000
C)$955; $21,100
D)$1000; $11,000

Accepted Answer

The amount Patricia deposited during her twentieth year can be found using the arithmetic sequence formula: $a_n = a_1 + (n-1)d$, where $a_n$ is the amount deposited in the nth year, $a_1$ is the initial amount deposited ($100), $n$ is the year number, and $d$ is the common difference between the deposits ($45). For the 20th year: $a_{20} = 100 + (20-1)45 = 100 + 855 = $955$. The total amount deposited over 20 years can be calculated using the formula for the sum of an arithmetic series: $S_n = \frac{n}{2}[2a_1 + (n-1)d]$, where $S_n$ is the total amount deposited over n years. For 20 years: $S_{20} = \frac{20}{2}[2(100) + (20-1)45] = 10[200 + 855] = 10[1055] = $10,550$.

Question 6

Write the first five terms of the geometric sequence with the given information.
-The first term is 4 and the common ratio is 3.
A)4, 7, 10, 13, 16
B)3, 12, 48, 192, 768
C)4, 12, 36, 108, 324
D)12, 36, 108, 324, 972

Accepted Answer

The first term is 4, and each subsequent term is found by multiplying the previous term by the common ratio of 3. Therefore, the sequence is 4, 12, 36, 108, 324.

Question 7

Solve the problem.
-John does 10 pushups on the first day of a 30-day month, and then increases the number of pushups by 3 pushups a day. How many pushups has he done by the end of the month?
A)1595 pushups
B)1605 pushups
C)1695 pushups
D)1650 pushups

Accepted Answer

John's pushups form an arithmetic sequence with the first term a1 = 10 and the common difference d = 3. The nth term is given by an = a1 + (n-1)d. For 30 days, the last term a30 = 10 + (30-1)*3 = 97. The sum of an arithmetic sequence is S = n/2 * (a1 + an). So, S = 30/2 * (10 + 97) = 15 * 107 = 1605 pushups.

Question 8

Solve the problem.
-Suppose you are offered a job with a starting salary of $37,000 per year and a guaranteed raise of $1500 per year. How much money will you make in total over the first 20 years?
A)$960,000
B)$1,025,000
C)$65,500
D)$1,124,600

Accepted Answer

This problem can be solved by calculating the total sum of an arithmetic series. The first term (a) is $37,000, the common difference (d) is $1,500, and the number of terms (n) is 20. The sum of an arithmetic series is given by the formula S = n/2[2a + (n-1)d]. Substituting the given values, we get S = 20/2[2(37,000) + (20-1)1,500] = 10[74,000 + 28,500] = 10[102,500] = $1,025,000.

Question 9

Write the first five terms of the geometric sequence with the given information.
-The first term is 8 and the common ratio is $- \frac { 1 } { 4 }$.
A) $- 2 , \frac { 1 } { 2 } , - \frac { 1 } { 8 } , \frac { 1 } { 32 } , - \frac { 1 } { 128 }$
B) $8 , - 32,128 , - 512,2048$
C) $8 , - 2 , \frac { 1 } { 2 } , - \frac { 1 } { 8 } , \frac { 1 } { 32 }$
D) $2 , - \frac { 1 } { 2 } , - \frac { 1 } { 8 } , - \frac { 1 } { 32 } , - \frac { 1 } { 128 }$

Accepted Answer

The first term is 8, and each subsequent term is found by multiplying the previous term by the common ratio $-\frac{1}{4}$. This results in the sequence $8, 8(-\frac{1}{4}), 8(-\frac{1}{4})^2, 8(-\frac{1}{4})^3, 8(-\frac{1}{4})^4$, which simplifies to $8, -2, \frac{1}{2}, -\frac{1}{8}, \frac{1}{32}$.

Question 10

Solve the problem.
-A new exhibit is scheduled to open at the local museum. Museum officials expect that 7000 people will visit the exhibit in its first week, and that the number of visitors will drop by 30 people per week after the first week during the first 6 months. Find the total number of visitors expected in the exhibit's first 7 weeks.
A)41,550 visitors
B)34,550 visitors
C)48,370 visitors
D)48,190 visitors

Accepted Answer

The answer of Solve the problem.
-A new exhibit is scheduled...

Question 11

Solve the problem.
-A certain club initially has 100 members and membership increases by 5 members every month. If dues are $5/month, what is the total amount of dues collected during the first year?
A)$7650
B)$6875
C)$7800
D)$7150

Accepted Answer

The club initially has 100 members and membership increases by 5 members every month. So, after 12 months, the club will have 100 + 5 * 12 = 160 members. The dues are $5/month, so the total amount of dues collected during the first year is 160 * 5 * 12 = $7650.

Question 12

Determine if the sequence is geometric. If the sequence is geometric, find the common ratio.
-4, -12, 36, -108, 324, . . .
A)yes, -3
B)no
C)yes, 12 $$\text { D) } y e s , - \frac { 1 } { 3 }$$

Accepted Answer

The sequence is geometric because each term is obtained by multiplying the previous term by -3.

Question 13

Write the first five terms of the geometric sequence with the given information.
-The first term is 7 and the common ratio is $\frac { 1 } { 2 }$.
A) $7,14,28,56,112$
B) $7 , \frac { 15 } { 2 } , 8 , \frac { 17 } { 2 } , 9$
C) $\frac { 7 } { 2 } , \frac { 7 } { 4 } , \frac { 7 } { 8 } , \frac { 7 } { 16 } , \frac { 7 } { 32 }$
D) $7 , \frac { 7 } { 2 } , \frac { 7 } { 4 } , \frac { 7 } { 8 } , \frac { 7 } { 16 }$

Accepted Answer

The sequence starts with 7 and each term is multiplied by $\frac { 1 } { 2 }$ to get the next term, resulting in $7, \frac { 7 } { 2 }, \frac { 7 } { 4 }, \frac { 7 } { 8 }, \frac { 7 } { 16 }$.

Question 14

Determine if the sequence is geometric. If the sequence is geometric, find the common ratio.
-6, -6, 6, -6, 6, . . .
A)yes, - 6
B)no
C)yes, -1
D)yes, 6

Accepted Answer

The sequence is geometric because each term is obtained by multiplying the previous term by -1.

Question 15

Write the first five terms of the geometric sequence with the given information.
-The first term is -5 and the common ratio is -4.
A)-4, 20, -80, 320, -1280
B)-5, -9, -13, -17, -21
C)-5, -20, -80, 320, -1280
D)-5, 20, -80, 320, -1280

Accepted Answer

The first term is -5, and each subsequent term is found by multiplying the previous term by the common ratio of -4. Thus, the sequence is -5, -5*(-4) = 20, 20*(-4) = -80, -80*(-4) = 320, 320*(-4) = -1280.

Question 16

Write the first five terms of the geometric sequence with the given information.
-$$a _ { n } = 7 a _ { n - 1 } ; a _ { 1 } = 4$$
A) $4,11,18,25,32$
B) $7,28,196,1372,9604$
C) $4,28,196,1372,9604$
D) $28,196,1372,9604,67,228$

Accepted Answer

The sequence is geometric with a common ratio of 7 and a first term of 4. Each term is found by multiplying the previous term by 7. The first five terms are 4, 28, 196, 1372, 9604.

Question 17

Write the first five terms of the geometric sequence with the given information.
-$$a _ { n } = - 2 a _ { n } - 1 ; a _ { 1 } = - 3$$
A) $6 , - 12,24 , - 48,96$
B) $- 2 , - 6 , - 12 , - 24 , - 48$
C) $- 3,6 , - 12,24 , - 48$
D) $- 3 , - 5 , - 7 , - 9 , - 11$

Accepted Answer

The given formula $a_n = -2a_{n-1}$ and initial value $a_1 = -3$ define a geometric sequence where each term is -2 times the previous term. Starting with -3, the sequence is $-3, 6, -12, 24, -48$.

Question 18

Determine if the sequence is geometric. If the sequence is geometric, find the common ratio.
-5.56, 8.38, 11.20, 14.02, . . .
A)no
B)yes, 2.82
C)yes, 2.12
D)yes, 8.46

Accepted Answer

The sequence is not geometric because the ratio between consecutive terms is not constant.

Question 19

Determine if the sequence is geometric. If the sequence is geometric, find the common ratio.
--7, -11, -15, -19, . . .
A)yes, -12
B)no
C)yes, -4
D)yes, 4

Accepted Answer

The sequence is not geometric because the difference between each term is constant (-4), indicating it is an arithmetic sequence, not geometric. In a geometric sequence, each term is a constant multiple of the previous term, not a constant difference.

Question 20

Determine if the sequence is geometric. If the sequence is geometric, find the common ratio.
--3, -12, -48, -192, -768, . . .
A)yes, -4
B)yes, 4
C)no $$\text { D) yes, } \frac { 1 } { 4 }$$

Accepted Answer

The answer of Determine if the sequence is geometric. If...

Quiz 9: Sequences and Series; Counting and Probability

Solve the problem. -A brick staircase has a total of 19 steps The first step requires 111 bricks. Each successive step requires 4 fewer bricks than the prior one. How many bricks are required to build the staircase?

Solve the problem. -A theater has 30 rows with 24 seats in the first row, 28 in the second row, 32 in the third row, and so forth. How many seats are in the theater?

Determine if the sequence is geometric. If the sequence is geometric, find the common ratio. -3, 12, 48, 192, 768, . . .

Solve the problem. -A person puts $20 into a bank account on January 1, $25 on February 1, $30 on March 1, and so forth. How much has the person put into the bank account by December 30?

Write the first five terms of the geometric sequence with the given information. -The first term is 4 and the common ratio is 3.

Solve the problem. -John does 10 pushups on the first day of a 30-day month, and then increases the number of pushups by 3 pushups a day. How many pushups has he done by the end of the month?

Solve the problem. -Suppose you are offered a job with a starting salary of $37,000 per year and a guaranteed raise of $1500 per year. How much money will you make in total over the first 20 years?

Write the first five terms of the geometric sequence with the given information. -The first term is 8 and the common ratio is $- \frac { 1 } { 4 }$ .

Solve the problem. -A certain club initially has 100 members and membership increases by 5 members every month. If dues are $5/month, what is the total amount of dues collected during the first year?

Determine if the sequence is geometric. If the sequence is geometric, find the common ratio. -4, -12, 36, -108, 324, . . .

Write the first five terms of the geometric sequence with the given information. -The first term is 7 and the common ratio is $\frac { 1 } { 2 }$ .

Determine if the sequence is geometric. If the sequence is geometric, find the common ratio. -6, -6, 6, -6, 6, . . .

Write the first five terms of the geometric sequence with the given information. -The first term is -5 and the common ratio is -4.

Write the first five terms of the geometric sequence with the given information. - $a _ { n } = 7 a _ { n - 1 } ; a _ { 1 } = 4$

Write the first five terms of the geometric sequence with the given information. - $a _ { n } = - 2 a _ { n } - 1 ; a _ { 1 } = - 3$

Determine if the sequence is geometric. If the sequence is geometric, find the common ratio. -5.56, 8.38, 11.20, 14.02, . . .

Determine if the sequence is geometric. If the sequence is geometric, find the common ratio. --7, -11, -15, -19, . . .

Determine if the sequence is geometric. If the sequence is geometric, find the common ratio. --3, -12, -48, -192, -768, . . .