Question 1

The annual sales $a _ { n }$ (in millions of dollars) for a certain company from 2001 to 2006 can be approximated by the model $a _ { n } = 500.7 + 161.52 n,$ $n = 0,1 , K , 5$ where $n$ represents the year, with $n = 0$ corresponding to 2001. Find the total sales from 2001 to 2004. Round to the nearest million.

Accepted Answer

The total sales from 2001 to 2004 can be calculated by summing the sales for each year from $n = 0$ to $n = 3$. The sales for each year are: $a_0 = 500.7$, $a_1 = 662.22$, $a_2 = 823.74$, $a_3 = 985.26$. Adding these gives approximately $2972 million.

Question 2

Find the sum of the integers from 5 to 27.

Accepted Answer

The answer of Find the sum of the integers from...

Question 3

Determine whether the sequence is geometric. If so, find the common ratio. 1, -3, 9, -27, ...

Accepted Answer

The sequence is geometric with a common ratio of -3, as each term is obtained by multiplying the previous term by -3.

Question 4

The seating section in a theater has 29 seats in the first row, 34 seats in the second row, and so on, increasing by 5 seats each row for a total of 15 rows. How many seats are in the thirteenth row?

Accepted Answer

The number of seats in each row forms an arithmetic sequence with the first term $a_1 = 29$ and a common difference $d = 5$. The $n$th term of an arithmetic sequence is given by $a_n = a_1 + (n - 1)d$. For the 13th row, $n = 13$, so $a_{13} = 29 + (13 - 1) \times 5 = 29 + 60 = 89$.

Question 5

Use summation notation to write the sum below. $2 + 4 + 6 + 8 + 10 + \mathrm { K } + 200$

Accepted Answer

The answer of Use summation notation to write the sum...

Question 6

Match the arithmetic sequence with its graph from the choices below. $a _ { n } = 35 - 3 n$

Accepted Answer

The answer of Match the arithmetic sequence with its graph...

Question 7

Find a formula for $a _ { n }$ for the arithmetic sequence below. $- 2 , - 1,0,1,2 , \mathbf { K }$

Accepted Answer

The given sequence is an arithmetic sequence with a common difference of 1 (each term increases by 1). The first term $a_1 = -2$. The formula for the $n$th term of an arithmetic sequence is $a_n = a_1 + (n - 1)d$, where $d$ is the common difference. Substituting $a_1 = -2$ and $d = 1$, we get $a_n = -2 + (n - 1)(1) = n - 3$.

Question 8

Find the indicated nth term of the geometric sequence. 7th term: $a _ { 5 } = \frac { 4 } { 81 } , a _ { 10 } = \frac { 4 } { 19,683 }$

Accepted Answer

The answer of Find the indicated nth term of the...

Question 9

Determine whether the sequence is geometric. If so, find the common ratio. 5, 7, 9, 11, ...

Accepted Answer

The sequence is not geometric because the ratio between consecutive terms is not constant. The differences between terms are constant (+2), indicating an arithmetic sequence, not a geometric one.

Question 10

Find the indicated nth partial sum of the arithmetic sequence. 3.4, 6.2, 9, 11.8, ..., n = 10

Accepted Answer

The answer of Find the indicated nth partial sum of...

Question 11

A heavy object (with negligible air resistance) is dropped from a plane. During the first second of fall, the object falls 17.4 meters; during the second second, it falls 52.2 meters; during the third second, it falls 87.0 meters; and during the fourth second, it falls 121.8 meters. If this pattern continues, how many meters will the object fall in 10 seconds?

Accepted Answer

The distance the object falls during each second increases by 34.8 meters each time (17.4 meters during the first second, 52.2 meters during the second second, etc.). Thus, during the tenth second, the object will fall 17.4 + 52.2 + 87.0 + 121.8 + 156.6 + 191.4 + 226.2 + 261.0 + 295.8 + 330.6 meters = 1740.0 meters.

Question 12

Find the partial sum. $\sum _ { n = 1 } ^ { 140 } ( - 4 n + 1 )$

Accepted Answer

The given sum can be simplified by distributing the summation over each term: $\sum _ { n = 1 } ^ { 140 } ( - 4 n + 1 ) = \sum _ { n = 1 } ^ { 140 } ( - 4 n ) + \sum _ { n = 1 } ^ { 140 } 1$. The first part is an arithmetic series with a common difference of -4, and the second part is simply adding 1 a total of 140 times. The sum of an arithmetic series can be found using the formula $S = \frac{n}{2} [2a + (n-1)d]$, where $n$ is the number of terms, $a$ is the first term, and $d$ is the common difference. For the first part, $a = -4$, $d = -4$, and $n = 140$, so the sum is $S_1 = \frac{140}{2} [2(-4) + (140-1)(-4)] = 70[-8 - 556] = 70[-564] = -39480$. The second part is simply $140 \times 1 = 140$. Adding these two results gives the total sum: $-39480 + 140 = -39340$.

Question 13

Match the geometric sequence with its graph from the choices below. $a _ { n } = 12 \left( - \frac { 4 } { 3 } \right) ^ { n - 1 }$

Accepted Answer

The answer of Match the geometric sequence with its graph...

Question 14

Use a graphing utility to graph the first 10 terms of the sequence. $a _ { n } = - 2 ( 0.6 ) ^ { n - 1 }$

Accepted Answer

The answer of Use a graphing utility to graph the...

Question 15

Find the nth term of the geometric sequence. $- 2 , - \frac { 5 } { 2 } , - \frac { 2.5 } { 8 } , \mathbf { K }$

Accepted Answer

The common ratio (r) can be found by dividing any term in the sequence by its preceding term, e.g., $-\frac{5}{2} / -2 = \frac{5}{4}$. The first term (a) is $-2$. Thus, the nth term formula for a geometric sequence is $a \cdot r^{n-1}$, which gives $-2 \left( \frac{5}{4} \right)^{n-1}$.

Question 16

Write the first five terms of the arithmetic sequence. $a _ { 5 } = 19 , a _ { 15 } = 79$

Accepted Answer

The common difference $d$ can be found using the formula for the $n$th term of an arithmetic sequence: $a_n = a_1 + (n - 1)d$. Given $a_5 = 19$ and $a_{15} = 79$, we can set up two equations: $a_1 + 4d = 19$ and $a_1 + 14d = 79$. Solving these equations simultaneously, we find $d = 6$ and $a_1 = -5$. Thus, the first five terms are -5, 1, 7, 13, 19.

Question 17

Write the first five terms of the geometric sequence. $a _ { 1 } = - 1 , r = - \frac { 1 } { 6 }$

Accepted Answer

The sequence starts with $a_1 = -1$ and has a common ratio $r = -\frac{1}{6}$. Each term is found by multiplying the previous term by $r$. The first five terms are $-1, -1 \cdot -\frac{1}{6}, -1 \cdot (-\frac{1}{6})^2, -1 \cdot (-\frac{1}{6})^3, -1 \cdot (-\frac{1}{6})^4$, simplifying to $-1, \frac{1}{6}, -\frac{1}{36}, \frac{1}{216}, -\frac{1}{1296}$.

Question 18

Find the nth term of the geometric sequence. $3 , - 12,48 , \mathrm {~K}$

Accepted Answer

The common ratio of the sequence is $-4$, so the nth term is given by $3(-4)^{n-1}$.

Question 19

Logs are stacked so that there are 17 logs in the bottom row, 16 logs in the second row from the bottom, and so on, decreasing by 1 log each row. How many logs are there in the first five rows from the bottom?

Accepted Answer

The number of logs in the first five rows is the sum of the first five terms of the sequence: 17, 16, 15, 14, 13. The sum is 17 + 16 + 15 + 14 + 13 = 75 logs.

Question 20

Consider a job offer with a starting salary of $43,200 and a given annual raise of $2175. Determine the total compensation from the company through seven full years of employment.

Accepted Answer

In one year, the salary will increase from $43,200 to $43,200 + $2,175 = $45,375.
Therefore, the total compensation in the first year is $45,375.
In the second year, the salary will increase from $45,375 to $45,375 + $2,175 = $47,550.
Therefore, the total compensation in the second year is $45,375 + $47,550 = $92,925.
Similarly, we can calculate the total compensation for each year:

Year 1: $45,375
Year 2: $92,925 = $45,375 + $47,550
Year 3: $141,075 = $92,925 + $48,150
Year 4: $189,225 = $141,075 + $48,150
Year 5: $237,375 = $189,225 + $48,150
Year 6: $285,525 = $237,375 + $48,150
Year 7: $333,675 = $285,525 + $48,150

Therefore, the total compensation through seven full years of employment is $333,675. The correct answer is E.

Quiz 16: Series and Taylor Polynomials Web

Find the sum of the integers from 5 to 27.

Determine whether the sequence is geometric. If so, find the common ratio. 1, -3, 9, -27, ...

The seating section in a theater has 29 seats in the first row, 34 seats in the second row, and so on, increasing by 5 seats each row for a total of 15 rows. How many seats are in the thirteenth row?

Use summation notation to write the sum below. 2+4+6+8+10+K+2002 + 4 + 6 + 8 + 10 + \mathrm { K } + 2002+4+6+8+10+K+200

Match the arithmetic sequence with its graph from the choices below. an=35−3na _ { n } = 35 - 3 nan​=35−3n

Find a formula for ana _ { n }an​ for the arithmetic sequence below. −2,−1,0,1,2,K- 2 , - 1,0,1,2 , \mathbf { K }−2,−1,0,1,2,K

Find the indicated nth term of the geometric sequence. 7th term: a5=481,a10=419,683a _ { 5 } = \frac { 4 } { 81 } , a _ { 10 } = \frac { 4 } { 19,683 }a5​=814​,a10​=19,6834​

Determine whether the sequence is geometric. If so, find the common ratio. 5, 7, 9, 11, ...

Find the indicated nth partial sum of the arithmetic sequence. 3.4, 6.2, 9, 11.8, ..., n = 10

Find the partial sum. ∑n=1140(−4n+1) \sum _ { n = 1 } ^ { 140 } ( - 4 n + 1 ) ∑n=1140​(−4n+1)

Match the geometric sequence with its graph from the choices below. an=12(−43) n−1a _ { n } = 12 \left( - \frac { 4 } { 3 } \right) ^ { n - 1 }an​=12(−34​) n−1

Use a graphing utility to graph the first 10 terms of the sequence. an=−2(0.6) n−1a _ { n } = - 2 ( 0.6 ) ^ { n - 1 }an​=−2(0.6) n−1

Find the nth term of the geometric sequence. −2,−52,−2.58,K- 2 , - \frac { 5 } { 2 } , - \frac { 2.5 } { 8 } , \mathbf { K }−2,−25​,−82.5​,K

Write the first five terms of the arithmetic sequence. a5=19,a15=79a _ { 5 } = 19 , a _ { 15 } = 79a5​=19,a15​=79

Write the first five terms of the geometric sequence. a1=−1,r=−16a _ { 1 } = - 1 , r = - \frac { 1 } { 6 }a1​=−1,r=−61​

Find the nth term of the geometric sequence. 3,−12,48, K3 , - 12,48 , \mathrm {~K}3,−12,48, K