Question 1

Write out the first three terms and the last term of the arithmetic sequence.
-$$\sum _ { i = 1 } ^ { 100 } - 3 i$$

Accepted Answer

The given sequence is an arithmetic sequence with the first term $a_1 = -3$ and a common difference $d = -3$. The $n$th term of an arithmetic sequence is given by $a_n = a_1 + (n - 1)d$. For $n = 100$, the last term is $a_{100} = -3 + (100 - 1)(-3) = -3 - 297 = -300$. The first three terms are $-3, -6, -9$, and the last term is $-300$, matching option A.

Question 2

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
-$$\sum _ { i = 1 } ^ { 39 } ( - 2 i - 1 )$$

Accepted Answer

The sum of the first n terms of an arithmetic sequence can be found using the formula $S_n = \frac{n}{2} [2a + (n - 1)d]$, where $n$ is the number of terms, $a$ is the first term, and $d$ is the common difference. Here, $a = -2(1) - 1 = -3$, $d = -2$, and $n = 39$. Plugging these values into the formula gives $S_{39} = \frac{39}{2} [2(-3) + (39 - 1)(-2)] = \frac{39}{2} [-6 - 76] = \frac{39}{2} [-82] = -1599$.

Question 3

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
-$$\sum _ { i = 1 } ^ { 50 } ( - 6 i + 8 )$$

Accepted Answer

The sum of the first n terms of an arithmetic sequence can be found using the formula $S_n = \frac{n}{2} [2a + (n - 1)d]$, where $S_n$ is the sum of the first n terms, $a$ is the first term, $n$ is the number of terms, and $d$ is the common difference. For the given sequence, the first term $a = -6(1) + 8 = 2$, and the common difference $d = -6$. Plugging these values into the formula with $n = 50$, we get $S_{50} = \frac{50}{2} [2(2) + (50 - 1)(-6)] = 25[4 - 294] = 25(-290) = -7250$.

Question 4

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
-$$\sum _ { \mathrm { i } = 1 } ^ { 27 } ( 2 \mathrm { i } - 5 )$$

Accepted Answer

The sum of an arithmetic sequence can be found using the formula $S_n = \frac{n}{2} [2a + (n - 1)d]$, where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $n$ is the number of terms, and $d$ is the common difference. Here, $a = 2(1) - 5 = -3$, $d = 2$, and $n = 27$. Plugging these values into the formula gives $S_{27} = \frac{27}{2} [2(-3) + (27 - 1)2] = \frac{27}{2} [-6 + 52] = \frac{27}{2} \times 46 = 621$.

Question 5

Write Terms of a Geometric Sequence
-$$\mathrm { a } _ { 1 } = 5 ; \mathrm { r } = 4$$

Accepted Answer

The terms of a geometric sequence are found by starting with the first term and multiplying by the common ratio $r$ for each subsequent term. Given $\mathrm { a } _ { 1 } = 5$ and $r = 4$, the sequence is $5, 5\cdot4, 5\cdot4^2, 5\cdot4^3, 5\cdot4^4$, which simplifies to $5, 20, 80, 320, 1280$.

Question 6

Solve the problem.
-A brick staircase has a total of 15 steps The bottom step requires 114 bricks. Each successive step requires 5 fewer bricks than the prior one. How many bricks are required to build the staircase?

Accepted Answer

The total number of bricks required for the staircase can be found using the formula for the sum of an arithmetic series, $S_n = \frac{n}{2}(a_1 + a_n)$, where $n$ is the number of terms, $a_1$ is the first term, and $a_n$ is the last term. Here, $n = 15$, $a_1 = 114$, and $a_n = 114 - 5(15 - 1) = 114 - 70 = 44$. Plugging these values into the formula gives $S_{15} = \frac{15}{2}(114 + 44) = 7.5 \times 158 = 1185$.

Question 7

Find the Common Ratio of a Geometric Sequence
If the given sequence is a geometric sequence, find the common ratio.
-$3 , - 9,27 , - 81,243$

Accepted Answer

The common ratio of a geometric sequence is found by dividing any term by its preceding term. For this sequence, dividing -9 by 3 gives -3, 27 by -9 also gives -3, and so on. This shows that the common ratio is -3.

Question 8

Solve the problem.
-A person puts $29 into a bank account on January 1, $39 on February 1, $49 on March 1, and so forth. How much has the person put into the bank account by December 30?

Accepted Answer

The person deposits an increasing amount each month, starting from $29 in January and increasing by $10 each month. The sequence of deposits is an arithmetic sequence with the first term $29 and the common difference $10. The 12th term (December) is $29 + 11*$10 = $139. The sum of the deposits is the sum of the first 12 terms of this arithmetic sequence, which is (12/2) * (29 + 139) = 6 * 168 = $1008.

Question 9

Solve the problem.
-A theater has 40 rows with 29 seats in the first row, 32 in the second row, 35 in the third row, and so forth. How many seats are in the theater?

Accepted Answer

The number of seats in each row increases by 3. So, the number of seats in the 40th row is 29 + (40 - 1) * 3 = 140. The total number of seats is the sum of the number of seats in each row, which is (29 + 140) * 40 / 2 = 3560.

Question 10

Find the Common Ratio of a Geometric Sequence
If the given sequence is a geometric sequence, find the common ratio.
-$$\frac { 2 } { 3 } , \frac { 8 } { 3 } , \frac { 32 } { 3 } , \frac { 128 } { 3 } , \frac { 512 } { 3 }$$

Accepted Answer

To find the common ratio, we divide each term by the one that comes before it. We get: 
$$\frac { 8/3 }{ 2/3 } = 4$$ 
$$\frac { 32/3 }{ 8/3 } = 4$$ 
$$\frac { 128/3 }{ 32/3 } = 4$$ 
$$\frac { 512/3 }{ 128/3 } = 4$$

Since each division gave us the same answer of 4, it is the common ratio.

Question 11

Find the Common Ratio of a Geometric Sequence
If the given sequence is a geometric sequence, find the common ratio.
-$$\frac { 3 } { 3 } , \frac { 3 } { 6 } , \frac { 3 } { 12 } , \frac { 3 } { 24 } , \frac { 3 } { 48 }$$

Accepted Answer

The common ratio of a geometric sequence is found by dividing any term by the previous term. For this sequence, dividing the second term $\frac{3}{6}$ by the first term $\frac{3}{3}$ gives $\frac{1}{2}$, which is the common ratio. This ratio remains consistent throughout the sequence.

Question 12

Write Terms of a Geometric Sequence
-$$\mathrm { a } _ { 1 } = 7 ; \mathrm { r } = - 3$$

Accepted Answer

The terms of a geometric sequence can be found using the formula $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number. Starting with $a_1 = 7$ and $r = -3$, the sequence is $7, 7 \cdot (-3) = -21, -21 \cdot (-3) = 63, 63 \cdot (-3) = -189, -189 \cdot (-3) = 567$.

Question 13

Write Terms of a Geometric Sequence
-$$a _ { 1 } = 2 ; r = \frac { 1 } { 3 }$$

Accepted Answer

The terms of a geometric sequence are found by starting with the first term and repeatedly multiplying by the common ratio. Here, $a_1 = 2$ and $r = \frac{1}{3}$, so each term is $\frac{1}{3}$ of the previous term, which matches option A.

Question 14

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
-$$\sum _ { i = 1 } ^ { 42 } ( 2 i + 3 )$$

Accepted Answer

The given sum can be simplified to find the sum of an arithmetic sequence. The formula for the sum of the first n terms of an arithmetic sequence is $S_n = \frac{n}{2} [2a + (n - 1)d]$, where $n$ is the number of terms, $a$ is the first term, and $d$ is the common difference. In this case, the first term when $i = 1$ is $2(1) + 3 = 5$, and the common difference $d$ is 2 (since each term increases by $2i$, and for each consecutive term, $i$ increases by 1, thus adding 2 to the sequence each time). Plugging into the formula: $S_{42} = \frac{42}{2} [2(5) + (42 - 1)2] = 21[10 + 82] = 21 \times 92 = 1932$.

Question 15

Write out the first three terms and the last term of the arithmetic sequence.
-$\sum _ { i = 1 } ^ { 17 } ( 4 i + 4 )$

Accepted Answer

The given sequence is an arithmetic sequence with the general term $4i + 4$. For the first term, $i = 1$, so the first term is $4(1) + 4 = 8$. The second term, with $i = 2$, is $4(2) + 4 = 12$, and the third term, with $i = 3$, is $4(3) + 4 = 16$. The last term, when $i = 17$, is $4(17) + 4 = 68 + 4 = 72$. Therefore, the sequence is $8 + 12 + 16 + \ldots + 72$.

Question 16

Solve the problem.
-A new exhibit is scheduled to open at the local museum. Museum officials expect that 6000 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.

Accepted Answer

The total number of visitors expected in the exhibit's first 7 weeks is 41,370.The number of visitors in the first week is 6000.The number of visitors in the second week is 6000 - 30 = 5970.The number of visitors in the third week is 5970 - 30 = 5940.The number of visitors in the fourth week is 5940 - 30 = 5910.The number of visitors in the fifth week is 5910 - 30 = 5880.The number of visitors in the sixth week is 5880 - 30 = 5850.The number of visitors in the seventh week is 5850 - 30 = 5820.The total number of visitors in the first 7 weeks is 6000 + 5970 + 5940 + 5910 + 5880 + 5850 + 5820 = 41,370.

Question 17

Find the Common Ratio of a Geometric Sequence
If the given sequence is a geometric sequence, find the common ratio.
-$$\frac { 1 } { 7 } , \frac { 1 } { 10 } , \frac { 1 } { 13 } , \frac { 1 } { 16 }$$

Accepted Answer

The given sequence does not have a constant ratio between consecutive terms, so it is not a geometric sequence.

Question 18

Find the Common Ratio of a Geometric Sequence
If the given sequence is a geometric sequence, find the common ratio.
-4, 8, 16, 32, 64

Accepted Answer

To find the common ratio of a geometric sequence, you divide any term by its preceding term. For example, 8/4 = 2 and 16/8 = 2. Therefore, the common ratio is 2.

Question 19

Solve the problem.
-As part of her retirement savings plan, Patricia deposited $150 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.

Accepted Answer

The amount deposited in the first year is $150. The amount deposited in the second year is $150 + $45 = $195. The amount deposited in the third year is $195 + $45 = $240. Continuing in this manner, the amount deposited in the twentieth year is $150 + $45(20 - 1) = $150 + $45(19) = $150 + $855 = $1005. The total amount deposited in the twenty years is $150 + $195 + $240 + ... + $1005 = $11,550.

Question 20

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
-$$\sum _ { i = 1 } ^ { 90 } - 5 i$$

Accepted Answer

The formula for the sum of the first $n$ terms of an arithmetic sequence is $S_n = \frac{n}{2}(a_1 + a_n)$, where $a_1$ is the first term and $a_n$ is the nth term. In this case, $a_1 = -5$ (since when $i = 1$, the term is $-5 \times 1 = -5$), and $a_n = -5 \times 90 = -450$. Plugging these values into the formula gives $S_{90} = \frac{90}{2}(-5 - 450) = 45 \times -455 = -20,475$.

Quiz 8: Sequences, Induction, and Probability

Write out the first three terms and the last term of the arithmetic sequence. - $\sum _ { i = 1 } ^ { 100 } - 3 i$

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - $\sum _ { i = 1 } ^ { 39 } ( - 2 i - 1 )$

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - $\sum _ { i = 1 } ^ { 50 } ( - 6 i + 8 )$

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - $\sum _ { \mathrm { i } = 1 } ^ { 27 } ( 2 \mathrm { i } - 5 )$

Write Terms of a Geometric Sequence - $\mathrm { a } _ { 1 } = 5 ; \mathrm { r } = 4$

Solve the problem. -A brick staircase has a total of 15 steps The bottom step requires 114 bricks. Each successive step requires 5 fewer bricks than the prior one. How many bricks are required to build the staircase?

Find the Common Ratio of a Geometric Sequence If the given sequence is a geometric sequence, find the common ratio. - $3 , - 9,27 , - 81,243$

Solve the problem. -A person puts $29 into a bank account on January 1, $39 on February 1, $49 on March 1, and so forth. How much has the person put into the bank account by December 30?

Solve the problem. -A theater has 40 rows with 29 seats in the first row, 32 in the second row, 35 in the third row, and so forth. How many seats are in the theater?

Find the Common Ratio of a Geometric Sequence If the given sequence is a geometric sequence, find the common ratio. - $\frac { 2 } { 3 } , \frac { 8 } { 3 } , \frac { 32 } { 3 } , \frac { 128 } { 3 } , \frac { 512 } { 3 }$

Find the Common Ratio of a Geometric Sequence If the given sequence is a geometric sequence, find the common ratio. - $\frac { 3 } { 3 } , \frac { 3 } { 6 } , \frac { 3 } { 12 } , \frac { 3 } { 24 } , \frac { 3 } { 48 }$

Write Terms of a Geometric Sequence - $\mathrm { a } _ { 1 } = 7 ; \mathrm { r } = - 3$

Write Terms of a Geometric Sequence - $a _ { 1 } = 2 ; r = \frac { 1 } { 3 }$

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - $\sum _ { i = 1 } ^ { 42 } ( 2 i + 3 )$

Write out the first three terms and the last term of the arithmetic sequence. - $\sum _ { i = 1 } ^ { 17 } ( 4 i + 4 )$

Find the Common Ratio of a Geometric Sequence If the given sequence is a geometric sequence, find the common ratio. - $\frac { 1 } { 7 } , \frac { 1 } { 10 } , \frac { 1 } { 13 } , \frac { 1 } { 16 }$

Find the Common Ratio of a Geometric Sequence If the given sequence is a geometric sequence, find the common ratio. -4, 8, 16, 32, 64

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - $\sum _ { i = 1 } ^ { 90 } - 5 i$

Quiz 8: Sequences, Induction, and Probability

Write out the first three terms and the last term of the arithmetic sequence. - ∑i=1100−3i\sum _ { i = 1 } ^ { 100 } - 3 i∑i=1100​−3i

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - ∑i=139(−2i−1) \sum _ { i = 1 } ^ { 39 } ( - 2 i - 1 ) ∑i=139​(−2i−1)

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - ∑i=150(−6i+8) \sum _ { i = 1 } ^ { 50 } ( - 6 i + 8 ) ∑i=150​(−6i+8)

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - ∑i=127(2i−5) \sum _ { \mathrm { i } = 1 } ^ { 27 } ( 2 \mathrm { i } - 5 ) ∑i=127​(2i−5)

Write Terms of a Geometric Sequence - a1=5;r=4\mathrm { a } _ { 1 } = 5 ; \mathrm { r } = 4a1​=5;r=4

Solve the problem. -A brick staircase has a total of 15 steps The bottom step requires 114 bricks. Each successive step requires 5 fewer bricks than the prior one. How many bricks are required to build the staircase?

Find the Common Ratio of a Geometric Sequence If the given sequence is a geometric sequence, find the common ratio. - 3,−9,27,−81,2433 , - 9,27 , - 81,2433,−9,27,−81,243

Solve the problem. -A person puts $29 into a bank account on January 1, $39 on February 1, $49 on March 1, and so forth. How much has the person put into the bank account by December 30?

Solve the problem. -A theater has 40 rows with 29 seats in the first row, 32 in the second row, 35 in the third row, and so forth. How many seats are in the theater?

Find the Common Ratio of a Geometric Sequence If the given sequence is a geometric sequence, find the common ratio. - 23,83,323,1283,5123\frac { 2 } { 3 } , \frac { 8 } { 3 } , \frac { 32 } { 3 } , \frac { 128 } { 3 } , \frac { 512 } { 3 }32​,38​,332​,3128​,3512​

Find the Common Ratio of a Geometric Sequence If the given sequence is a geometric sequence, find the common ratio. - 33,36,312,324,348\frac { 3 } { 3 } , \frac { 3 } { 6 } , \frac { 3 } { 12 } , \frac { 3 } { 24 } , \frac { 3 } { 48 }33​,63​,123​,243​,483​

Write Terms of a Geometric Sequence - a1=7;r=−3\mathrm { a } _ { 1 } = 7 ; \mathrm { r } = - 3a1​=7;r=−3

Write Terms of a Geometric Sequence - a1=2;r=13a _ { 1 } = 2 ; r = \frac { 1 } { 3 }a1​=2;r=31​

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum. - ∑i=142(2i+3) \sum _ { i = 1 } ^ { 42 } ( 2 i + 3 ) ∑i=142​(2i+3)

Write out the first three terms and the last term of the arithmetic sequence. - ∑i=117(4i+4) \sum _ { i = 1 } ^ { 17 } ( 4 i + 4 ) ∑i=117​(4i+4)

Find the Common Ratio of a Geometric Sequence If the given sequence is a geometric sequence, find the common ratio. - 17,110,113,116\frac { 1 } { 7 } , \frac { 1 } { 10 } , \frac { 1 } { 13 } , \frac { 1 } { 16 }71​,101​,131​,161​