Question 1

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The difference between consecutive terms in an arithmetic sequence is called the________ ________, and is denoted by d.

Accepted Answer

The answer of Write the word or phrase that best...

Question 2

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-An arithmetic sequence is a linear function whose domain is the set of ________integers.

Accepted Answer

The answer of Write the word or phrase that best...

Question 3

Write the sum using summation notation.
-$$\frac { x ^ { 2 } } { 1 } + \frac { x ^ { 4 } } { 2 } + \frac { x ^ { 6 } } { 6 } + \frac { x ^ { 8 } } { 24 } + \frac { x ^ { 10 } } { 120 }$$
A) $\sum _ { i = 1 } ^ { 5 } \frac { x ^ { 2 i } } { i ! }$
B) $\sum _ { i = 0 } ^ { 4 } \frac { x ^ { 2 i } } { i ! }$
C) $\sum _ { i = 1 } ^ { 5 } \frac { x ^ { 2 i } } { 2 ^ { i - 1 } }$
D) $\sum _ { i = 0 } ^ { 4 } \frac { x ^ { 2 i } } { ( i + 1 ) ! }$

Accepted Answer

The given series can be represented by the summation notation where each term is of the form $\frac{x^{2i}}{i!}$, starting from $i=1$ to $i=5$, matching the pattern of the series provided.

Question 4

Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
-In a business meeting, every person at the meeting shakes every other person's hand exactly one time. The total number of handshakes for n people at the meeting is given by $$a _ { n } = \frac { 1 } { 2 } n ( n - 1 )$$ Evaluate $a _ { 12 }$ and interpret its meaning in the context of this problem.
A) 132; If 12 people are present at the meeting there will be 132 handshakes.
B) 66; If 12 people are present at the meeting there will be 66 handshakes.
C) 11; The 12th person shakes performs 11 handshakes.
D) 78; If 12 people are present at the meeting there will be 78 handshakes.

Accepted Answer

The formula $\frac{1}{2} n (n - 1)$ calculates the number of unique handshakes. For 12 people, it evaluates to $\frac{1}{2} \times 12 \times 11 = 66$ handshakes.

Question 5

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-An________ sequence is a sequence in which each term after the first differs from its predecessor by a fixed constant.

Accepted Answer

The answer of Write the word or phrase that best...

Question 6

Choose the one alternative that best completes the statement or answers the question.
Determine whether the sequence is arithmetic. If so, find the common difference.
-4, -16, 64, -256, . . .
A) not arithmetic
B) arithmetic; d = -4
C) arithmetic; d = 4
D) arithmetic; d = -2

Accepted Answer

The sequence is not arithmetic because the difference between consecutive terms does not remain constant. Instead, each term is multiplied by -4 to get the next term, indicating a geometric sequence, not an arithmetic one.

Question 7

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-Given an arithmetic sequence with first term $a _ { 1 }$ , and common difference d, the nth term is
represented by the formula $a _ { \mathrm { n } } =$ ________.

Accepted Answer

The answer of Write the word or phrase that best...

Question 8

Write the first four terms of an arithmetic sequence, $\left\{ a _ { n } \right\}$ , based on the given information about the
sequence.
-$$a _ { 1 } = 3 , d = 6$$
A) $18,108,648,3,888$
B) $3,9,15,21$
C) $3,18,108,648$
D) $9,15,21,27$

Accepted Answer

The first term is given as $a_1 = 3$, and the common difference $d = 6$. The nth term of an arithmetic sequence is given by $a_n = a_1 + (n-1)d$. Thus, the first four terms are $3, 3+6=9, 9+6=15, 15+6=21$.

Question 9

Write the sum using summation notation.
-$$\frac { 1 } { x + 3 } + \frac { 2 } { x + 6 } + \frac { 6 } { x + 9 } + \frac { 24 } { x + 12 } + \frac { 120 } { x + 15 }$$
A) $\sum _ { i = 1 } ^ { 5 } \frac { i ! } { x + 3 i }$
B) $\sum _ { i = 1 } ^ { 5 } \frac { i ! } { x + i }$
C) $\sum _ { i = 1 } ^ { 5 } \frac { 2 ^ { i - 1 } } { x + 3 i }$
D) $\sum _ { i = 1 } ^ { 5 } \frac { i ! } { x + 3 }$

Accepted Answer

The terms in the series can be represented by $\frac{i!}{x + 3i}$, where $i!$ denotes the factorial of $i$ (e.g., $1!, 2!, 6!, 24!, 120!$), and the denominator increases by 3 for each term ($x+3, x+6, x+9, \ldots$). This matches the pattern described in option A.

Question 10

Choose the one alternative that best completes the statement or answers the question.
Determine whether the sequence is arithmetic. If so, find the common difference.
-$5 , \frac { 25 } { 6 } , \frac { 125 } { 36 } , \frac { 625 } { 216 }$
A) arithmetic; $d = 5$
B) not arithmetic
C) arithmetic; $d = \frac { 5 } { 6 }$
D) arithmetic; $d = \frac { 6 } { 5 }$

Accepted Answer

The sequence is not arithmetic because the difference between consecutive terms is not constant.

Question 11

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-Given an arithmetic sequence with first term a1 and nth term an, the nth partial sum is given by the
formula $S_{n}=$_______

Accepted Answer

The answer of Write the word or phrase that best...

Question 12

Write the sum using summation notation.
-$$- \frac { 1 } { 3 } + \frac { 1 } { 9 } - \frac { 1 } { 27 } + \frac { 1 } { 81 } - \frac { 1 } { 243 }$$
A) $\sum _ { i = 1 } ^ { 5 } \left( ( - 1 ) ^ { i + 1 } \frac { 1 } { 3 i } \right)$
B) $\sum _ { i = 1 } ^ { 5 } \left( ( - 1 ) ^ { i } \frac { 1 } { 3 i } \right)$
C) $\sum _ { i = 1 } ^ { 5 } \left( ( - 1 ) ^ { i } \frac { 1 } { 3 ^ { i } } \right)$
D) $\sum _ { i = 1 } ^ { 5 } \left( ( - 1 ) ^ { i + 1 } \frac { 1 } { 3 ^ { i } } \right)$

Accepted Answer

The answer of Write the sum using summation notation.
-\[- \frac...

Question 13

Write the first four terms of the arithmetic sequence with the given first term and common difference.
-$$a _ { 1 } = 9 , d = - 7$$
A) $9,16,23,30$
B) $9,2 , - 5 , - 12$
C) $- 7 , - 16 , - 25 , - 34$
D) $- 7,2,11,20$

Accepted Answer

The first term is given as 9, and each subsequent term is found by adding the common difference, which is -7, to the preceding term. Thus, the sequence is 9, 9-7=2, 2-7=-5, -5-7=-12.

Question 14

Solve the problem.
-Expenses for a company for year 1 are $\$ 20,000$. Every year thereafter, expenses increase by $\$ 1,500$ plus $2 \%$ of the cost of the prior year. Let $a _ { 1 }$ represent the original cost for year 1 ; that is $a _ { 1 } = 20,000$.
Use a recursive formula to find the cost $a _ { n }$ in terms of $a _ { n - 1 }$ for each subsequent year, $\mathrm { n } \geq 2$.
A) $a _ { n } = 1.02 a _ { n - 1 } + 21,500 , n \geq 2$
B) $a _ { n } = 0.02 a _ { n - 1 } + 1,500 , n \geq 2$
C) $a _ { n } = 1.02 a _ { n - 1 } + 1,500 , n \geq 2$
D) $a _ { n } = 0.02 a _ { n - 1 } + 21,500 , n \geq 2$

Accepted Answer

The correct formula is $a _ { n } = 1.02 a _ { n - 1 } + 1,500$, because each year the cost increases by $2\%$ (which is represented by multiplying the previous year's cost by $1.02$) and an additional $\$1,500$.

Question 15

Write the sum using summation notation.
-$$- 3 + 9 - 27 + 81$$
A) $\sum _ { n = 1 } ^ { 4 } ( - 1 ) ^ { n _ { 3 } n }$
B)$\sum_{n=1}^{4}(-1)^{n+1} 3^{n}$
C) $\sum _ { n = 1 } ^ { 4 } ( - 1 ) ^ { n + 1 } n$
D) $\sum _ { n = 1 } ^ { 4 } ( - 1 ) ^ { n _ { n } }$

Accepted Answer

The answer of Write the sum using summation notation.
-\[- 3...

Question 16

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The sum of the first n terms of a sequence is called the nth ________ sum and is denoted by $S _ { n }$

Accepted Answer

The answer of Write the word or phrase that best...

Question 17

Rewrite the series as an equivalent series with the new index of summation.
-$$\sum _ { i = 1 } ^ { 7 } ( 9 i ) = \sum _ { k = 3 } ^ { ? } ( ? )$$
A) $\sum _ { k = 3 } ^ { 9 } [ 9 ( k - 2 ) ]$
B) $\sum _ { k = 3 } ^ { 9 } [ 9 ( k + 2 ) ]$
C) $\sum _ { k = 3 } ^ { 9 } [ 9 ( k - 3 ) ]$
D) $\sum _ { k = 3 } ^ { 9 } [ 9 ( k + 3 ) ]$

Accepted Answer

To change the index of summation from $i = 1$ to $k = 3$, we substitute $i = k - 2$. This changes the expression inside the sum to $9(k - 2)$. The upper limit changes from $i = 7$ to $k = 9$ to maintain the same number of terms in the series.

Question 18

Write the sum using summation notation.
-$$a + a r + a r ^ { 2 } + a r ^ { 3 } + \ldots + a r ^ { 19 }$$
A) $\sum _ { i = 1 } ^ { 19 } a r ^ { j - 1 }$
B) $\sum _ { i = 1 } ^ { 20 } a r ^ { i - 1 }$
C) $\sum _ { i = 1 } ^ { 19 } a r ^ { i }$
D) $\sum _ { i = 1 } ^ { 20 } a r ^ { i }$

Accepted Answer

The series starts with $a$, which can be represented as $a r^{0}$, and goes up to $a r^{19}$. The summation notation that correctly represents this is $\sum _ { i = 1 } ^ { 20 } a r ^ { i - 1 }$, where $i$ starts from 1 and goes to 20, and $i-1$ correctly adjusts the exponent to start from 0 up to 19.

Question 19

Use the provided sums to evaluate the given expression.
-$\sum _ { i = 1 } ^ { 40 } i = 820$ and $\sum _ { i = 1 } ^ { 40 } i ^ { 2 } = 22,140 ;$ evaluate $\sum _ { i = 1 } ^ { 40 } \left( 5 i ^ { 2 } + 9 i \right)$
A) 118,080
B) 22,960
C) 111,520
D) 203,360

Accepted Answer

The given expression can be rewritten using the distributive property of summation as $5\sum _ { i = 1 } ^ { 40 } i ^ { 2 } + 9\sum _ { i = 1 } ^ { 40 } i$. Substituting the given sums, we get $5(22,140) + 9(820) = 110,700 + 7,380 = 118,080$.

Question 20

Write the first four terms of an arithmetic sequence, $\left\{ a _ { n } \right\}$ , based on the given information about the
sequence.
-Write the first five terms of the arithmetic sequence. $$a _ { 1 } = 3 , d = - 4$$
A) $- 4 , - 1,5,23,77$
B) $- 4 , - 1,2,5,8$
C) $3 , - 1,19 , - 61,259$
D) $3 , - 1 , - 5 , - 9 , - 13$

Accepted Answer

The first term is given as 3, and the common difference is -4. To find the next terms, subtract 4 from the previous term: $3 - 4 = -1$, $-1 - 4 = -5$, $-5 - 4 = -9$, and $-9 - 4 = -13$.

Quiz 8: Sequences, Series, Induction, and Probability

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The difference between consecutive terms in an arithmetic sequence is called the , and is denoted by d.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -An arithmetic sequence is a linear function whose domain is the set of ________integers.

Write the sum using summation notation. - $\frac { x ^ { 2 } } { 1 } + \frac { x ^ { 4 } } { 2 } + \frac { x ^ { 6 } } { 6 } + \frac { x ^ { 8 } } { 24 } + \frac { x ^ { 10 } } { 120 }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -An________ sequence is a sequence in which each term after the first differs from its predecessor by a fixed constant.

Choose the one alternative that best completes the statement or answers the question. Determine whether the sequence is arithmetic. If so, find the common difference. -4, -16, 64, -256, . . .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given an arithmetic sequence with first term $a _ { 1 }$ , and common difference d, the nth term is represented by the formula $a _ { \mathrm { n } } =$ ________.

Write the first four terms of an arithmetic sequence, $\left\{ a _ { n } \right\}$ , based on the given information about the sequence. - $a _ { 1 } = 3 , d = 6$

Write the sum using summation notation. - $\frac { 1 } { x + 3 } + \frac { 2 } { x + 6 } + \frac { 6 } { x + 9 } + \frac { 24 } { x + 12 } + \frac { 120 } { x + 15 }$

Choose the one alternative that best completes the statement or answers the question. Determine whether the sequence is arithmetic. If so, find the common difference. - $5 , \frac { 25 } { 6 } , \frac { 125 } { 36 } , \frac { 625 } { 216 }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given an arithmetic sequence with first term a1 and nth term an, the nth partial sum is given by the formula $S_{n}=$ _______

Write the sum using summation notation. - $- \frac { 1 } { 3 } + \frac { 1 } { 9 } - \frac { 1 } { 27 } + \frac { 1 } { 81 } - \frac { 1 } { 243 }$

Write the first four terms of the arithmetic sequence with the given first term and common difference. - $a _ { 1 } = 9 , d = - 7$

Write the sum using summation notation. - $- 3 + 9 - 27 + 81$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The sum of the first n terms of a sequence is called the nth ________ sum and is denoted by $S _ { n }$

Rewrite the series as an equivalent series with the new index of summation. - $\sum _ { i = 1 } ^ { 7 } ( 9 i ) = \sum _ { k = 3 } ^ { ? } ( ? )$

Write the sum using summation notation. - $a + a r + a r ^ { 2 } + a r ^ { 3 } + \ldots + a r ^ { 19 }$

Use the provided sums to evaluate the given expression. - $\sum _ { i = 1 } ^ { 40 } i = 820$ and $\sum _ { i = 1 } ^ { 40 } i ^ { 2 } = 22,140 ;$ evaluate $\sum _ { i = 1 } ^ { 40 } \left( 5 i ^ { 2 } + 9 i \right)$

Write the first four terms of an arithmetic sequence, $\left\{ a _ { n } \right\}$ , based on the given information about the sequence. -Write the first five terms of the arithmetic sequence. $a _ { 1 } = 3 , d = - 4$

Quiz 8: Sequences, Series, Induction, and Probability

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The difference between consecutive terms in an arithmetic sequence is called the________ ________, and is denoted by d.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -An arithmetic sequence is a linear function whose domain is the set of ________integers.

Write the sum using summation notation. - x21+x42+x66+x824+x10120\frac { x ^ { 2 } } { 1 } + \frac { x ^ { 4 } } { 2 } + \frac { x ^ { 6 } } { 6 } + \frac { x ^ { 8 } } { 24 } + \frac { x ^ { 10 } } { 120 }1x2​+2x4​+6x6​+24x8​+120x10​

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -An________ sequence is a sequence in which each term after the first differs from its predecessor by a fixed constant.

Choose the one alternative that best completes the statement or answers the question. Determine whether the sequence is arithmetic. If so, find the common difference. -4, -16, 64, -256, . . .

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given an arithmetic sequence with first term a1a _ { 1 }a1​ , and common difference d, the nth term is represented by the formula an=a _ { \mathrm { n } } =an​= ________.

Write the first four terms of an arithmetic sequence, {an}\left\{ a _ { n } \right\}{an​} , based on the given information about the sequence. - a1=3,d=6a _ { 1 } = 3 , d = 6a1​=3,d=6

Write the sum using summation notation. - 1x+3+2x+6+6x+9+24x+12+120x+15\frac { 1 } { x + 3 } + \frac { 2 } { x + 6 } + \frac { 6 } { x + 9 } + \frac { 24 } { x + 12 } + \frac { 120 } { x + 15 }x+31​+x+62​+x+96​+x+1224​+x+15120​

Choose the one alternative that best completes the statement or answers the question. Determine whether the sequence is arithmetic. If so, find the common difference. - 5,256,12536,6252165 , \frac { 25 } { 6 } , \frac { 125 } { 36 } , \frac { 625 } { 216 }5,625​,36125​,216625​

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given an arithmetic sequence with first term a1 and nth term an, the nth partial sum is given by the formula Sn=S_{n}=Sn​= _______

Write the sum using summation notation. - −13+19−127+181−1243- \frac { 1 } { 3 } + \frac { 1 } { 9 } - \frac { 1 } { 27 } + \frac { 1 } { 81 } - \frac { 1 } { 243 }−31​+91​−271​+811​−2431​

Write the first four terms of the arithmetic sequence with the given first term and common difference. - a1=9,d=−7a _ { 1 } = 9 , d = - 7a1​=9,d=−7

Write the sum using summation notation. - −3+9−27+81- 3 + 9 - 27 + 81−3+9−27+81

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The sum of the first n terms of a sequence is called the nth ________ sum and is denoted by SnS _ { n }Sn​

Rewrite the series as an equivalent series with the new index of summation. - ∑i=17(9i) =∑k=3?(?) \sum _ { i = 1 } ^ { 7 } ( 9 i ) = \sum _ { k = 3 } ^ { ? } ( ? ) ∑i=17​(9i) =∑k=3?​(?)

Write the sum using summation notation. - a+ar+ar2+ar3+…+ar19a + a r + a r ^ { 2 } + a r ^ { 3 } + \ldots + a r ^ { 19 }a+ar+ar2+ar3+…+ar19

Write the first four terms of an arithmetic sequence, {an}\left\{ a _ { n } \right\}{an​} , based on the given information about the sequence. -Write the first five terms of the arithmetic sequence. a1=3,d=−4a _ { 1 } = 3 , d = - 4a1​=3,d=−4

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The difference between consecutive terms in an arithmetic sequence is called the , and is denoted by d.

Write the sum using summation notation. - $\frac { x ^ { 2 } } { 1 } + \frac { x ^ { 4 } } { 2 } + \frac { x ^ { 6 } } { 6 } + \frac { x ^ { 8 } } { 24 } + \frac { x ^ { 10 } } { 120 }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given an arithmetic sequence with first term $a _ { 1 }$ , and common difference d, the nth term is represented by the formula $a _ { \mathrm { n } } =$ ________.

Write the first four terms of an arithmetic sequence, $\left\{ a _ { n } \right\}$ , based on the given information about the sequence. - $a _ { 1 } = 3 , d = 6$

Write the sum using summation notation. - $\frac { 1 } { x + 3 } + \frac { 2 } { x + 6 } + \frac { 6 } { x + 9 } + \frac { 24 } { x + 12 } + \frac { 120 } { x + 15 }$

Choose the one alternative that best completes the statement or answers the question. Determine whether the sequence is arithmetic. If so, find the common difference. - $5 , \frac { 25 } { 6 } , \frac { 125 } { 36 } , \frac { 625 } { 216 }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given an arithmetic sequence with first term a1 and nth term an, the nth partial sum is given by the formula $S_{n}=$ _______

Write the sum using summation notation. - $- \frac { 1 } { 3 } + \frac { 1 } { 9 } - \frac { 1 } { 27 } + \frac { 1 } { 81 } - \frac { 1 } { 243 }$

Write the first four terms of the arithmetic sequence with the given first term and common difference. - $a _ { 1 } = 9 , d = - 7$

Write the sum using summation notation. - $- 3 + 9 - 27 + 81$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The sum of the first n terms of a sequence is called the nth ________ sum and is denoted by $S _ { n }$

Rewrite the series as an equivalent series with the new index of summation. - $\sum _ { i = 1 } ^ { 7 } ( 9 i ) = \sum _ { k = 3 } ^ { ? } ( ? )$

Write the sum using summation notation. - $a + a r + a r ^ { 2 } + a r ^ { 3 } + \ldots + a r ^ { 19 }$

Write the first four terms of an arithmetic sequence, $\left\{ a _ { n } \right\}$ , based on the given information about the sequence. -Write the first five terms of the arithmetic sequence. $a _ { 1 } = 3 , d = - 4$