Precalculus

Problem 5

Describe the characteristics of a recursive sequence and give one example.

Amy Jiang

Numerade Educator

00:53

Problem 6

Describe the characteristics of an alternating sequence and give one example.

Amy Jiang

Numerade Educator

01:04

Problem 7

Find the first four terms, then find the 8 th and 12 th term for each $n$ th term given.
$$a_{n}=2 n-1$$

Amy Jiang

Numerade Educator

Problem 8

Find the first four terms, then find the 8 th and 12 th term for each $n$ th term given.
$$a_{n}=2 n+3$$

Amy Jiang

Numerade Educator

01:42

Problem 9

Find the first four terms, then find the 8 th and 12 th term for each $n$ th term given.
$$a_{n}=3 n^{2}-3$$

Amy Jiang

Numerade Educator

01:43

Problem 10

Find the first four terms, then find the 8 th and 12 th term for each $n$ th term given.
$$a_{n}=2 n^{3}-12$$

Amy Jiang

Numerade Educator

01:11

Problem 11

Find the first four terms, then find the 8 th and 12 th term for each $n$ th term given.
$$a_{n}=(-1)^{n} n$$

Amy Jiang

Numerade Educator

00:55

Problem 12

Find the first four terms, then find the 8 th and 12 th term for each $n$ th term given.
$$a_{n}=\frac{(-1)^{n}}{n}$$

Amy Jiang

Numerade Educator

00:58

Problem 13

Find the first four terms, then find the 8 th and 12 th term for each $n$ th term given.
$$a_{n}=\frac{n}{n+1}$$

Amy Jiang

Numerade Educator

02:25

Problem 14

Find the first four terms, then find the 8 th and 12 th term for each $n$ th term given.
$$a_{n}=\left(1+\frac{1}{n}\right)^{n}$$

Amy Jiang

Numerade Educator

01:03

Problem 15

Find the first four terms, then find the 8 th and 12 th term for each $n$ th term given.
$$a_{n}=\left(\frac{1}{2}\right)^{n}$$

Amy Jiang

Numerade Educator

01:16

Problem 16

Find the first four terms, then find the 8 th and 12 th term for each $n$ th term given.
$$a_{n}=\left(\frac{2}{3}\right)^{n}$$

Amy Jiang

Numerade Educator

00:28

Problem 17

Find the first four terms, then find the 8 th and 12 th term for each $n$ th term given.
$$a_{n}=\frac{1}{n}$$

Amy Jiang

Numerade Educator

00:43

Problem 18

Find the first four terms, then find the 8 th and 12 th term for each $n$ th term given.
$$a_{n}=\frac{1}{n^{2}}$$

Amy Jiang

Numerade Educator

01:35

Problem 19

Find the first four terms, then find the 8 th and 12 th term for each $n$ th term given.
$$a_{n}=\frac{(-1)^{n}}{n(n+1)}$$

Amy Jiang

Numerade Educator

Problem 20

Find the first four terms, then find the 8 th and 12 th term for each $n$ th term given.
$$a_{n}=\frac{(-1)^{n+1}}{2 n^{2}-1}$$

Amy Jiang

Numerade Educator

01:21

Problem 21

Find the first four terms, then find the 8 th and 12 th term for each $n$ th term given.
$$a_{n}=(-1)^{n} 2^{n}$$

Amy Jiang

Numerade Educator

01:24

Problem 22

Find the first four terms, then find the 8 th and 12 th term for each $n$ th term given.
$$a_{n}=(-1)^{n} 2^{-n}$$

Amy Jiang

Numerade Educator

00:21

Problem 23

Find the indicated term for each sequence.
$$a_{n}=n^{2}-2 ; a_{9}$$

Amy Jiang

Numerade Educator

00:19

Problem 24

Find the indicated term for each sequence.
$$a_{n}=(n-2)^{2} ; a_{9}$$

Amy Jiang

Numerade Educator

00:27

Problem 25

Find the indicated term for each sequence.
$$a_{n}=\frac{(-1)^{n+1}}{n} ; a_{5}$$

Amy Jiang

Numerade Educator

00:27

Problem 26

Find the indicated term for each sequence.
$$a_{n}=\frac{(-1)^{n+1}}{2 n-1} ; a_{5}$$

Amy Jiang

Numerade Educator

00:29

Problem 27

Find the indicated term for each sequence.
$$a_{n}=2\left(\frac{1}{2}\right)^{n-1} ; a_{7}$$

Amy Jiang

Numerade Educator

00:29

Problem 28

Find the indicated term for each sequence.
$$a_{n}=3\left(\frac{1}{3}\right)^{n-1} ; a_{7}$$

Amy Jiang

Numerade Educator

00:47

Problem 29

Find the indicated term for each sequence.
$$a_{n}=\left(1+\frac{1}{n}\right)^{n} ; a_{10}$$

Amy Jiang

Numerade Educator

00:52

Problem 30

Find the indicated term for each sequence.
$$\text { 30. } a_{n}=\left(n+\frac{1}{n}\right)^{n} ; a_{9}$$

Amy Jiang

Numerade Educator

00:23

Problem 31

Find the indicated term for each sequence.
$$a_{n}=\frac{1}{n(2 n+1)} ; a_{4}$$

Amy Jiang

Numerade Educator

00:34

Problem 32

Find the indicated term for each sequence.
$$a_{n}=\frac{1}{(2 n-1)(2 n+1)} ; a_{5}$$

Amy Jiang

Numerade Educator

01:30

Problem 33

Find the first five terms of each recursive sequence.
$$\left\{\begin{array}{l}a_{1}=2 \\a_{n}=5 a_{n-1}-3\end{array}\right.$$

Amy Jiang

Numerade Educator

Problem 34

Find the first five terms of each recursive sequence.
$$\left\{\begin{array}{l}a_{1}=3 \\a_{n}=2 a_{n-1}-3\end{array}\right.$$

Amy Jiang

Numerade Educator

Problem 35

Find the first five terms of each recursive sequence.
$$\left\{\begin{array}{l}a_{1}=-1 \\a_{n}=\left(a_{n-1}\right)^{2}+3\end{array}\right.$$

Amy Jiang

Numerade Educator

00:57

Problem 36

Find the first five terms of each recursive sequence.
$$\left\{\begin{array}{l}a_{1}=-2 \\a_{n}=a_{n-1}-16\end{array}\right.$$

Amy Jiang

Numerade Educator

01:20

Problem 37

Find the first five terms of each recursive sequence.
$$\left\{\begin{array}{l}c_{1}=64, c_{2}=32 \\c_{n}=\frac{c_{n-2}-c_{n-1}}{2}\end{array}\right.$$

Amy Jiang

Numerade Educator

01:22

Problem 38

Find the first five terms of each recursive sequence.
$$\left\{\begin{array}{l}c_{1}=1, c_{2}=2 \\c_{n}=c_{n-1}+\left(c_{n-2}\right)^{2}\end{array}\right.$$

Amy Jiang

Numerade Educator

00:45

Problem 39

Simplify each factorial expression.
$$\frac{8 !}{5 !}$$

Amy Jiang

Numerade Educator

00:28

Problem 40

Simplify each factorial expression.
$$\frac{12 !}{10 !}$$

Amy Jiang

Numerade Educator

00:38

Problem 41

Simplify each factorial expression.
$$\frac{9 !}{7 ! 2 !}$$

Amy Jiang

Numerade Educator

00:35

Problem 42

Simplify each factorial expression.
$$\frac{6 !}{3 ! 3 !}$$

Amy Jiang

Numerade Educator

00:38

Problem 43

Simplify each factorial expression.
$$\frac{8 !}{2 ! 6 !}$$

Amy Jiang

Numerade Educator

00:55

Problem 44

Simplify each factorial expression.
$$\frac{10 !}{3 ! 7 !}$$

Amy Jiang

Numerade Educator

Problem 45

Write out the first four terms in each sequence.
$$a_{n}=\frac{n !}{(n+1) !}$$

Amy Jiang

Numerade Educator

01:58

Problem 46

Write out the first four terms in each sequence.
$$a_{n}=\frac{n !}{(n+3) !}$$

Amy Jiang

Numerade Educator

01:59

Problem 47

Write out the first four terms in each sequence.
$$a_{n}=\frac{(n+1) !}{(3 n) !}$$

Amy Jiang

Numerade Educator

Problem 48

Write out the first four terms in each sequence.
$$a_{n}=\frac{(n+3) !}{(2 n) !}$$

Amy Jiang

Numerade Educator

01:08

Problem 49

Write out the first four terms in each sequence.
$$a_{n}=\frac{n^{n}}{n !}$$

Amy Jiang

Numerade Educator

01:01

Problem 50

Write out the first four terms in each sequence.
$$a_{n}=\frac{2^{n}}{n !}$$

Amy Jiang

Numerade Educator

00:31

Problem 51

Find the indicated partial sum for each sequence.
$$a_{n}=n ; S_{5}$$

Amy Jiang

Numerade Educator

01:06

Problem 52

Find the indicated partial sum for each sequence.
$$a_{n}=n^{2} ; S_{7}$$

Amy Jiang

Numerade Educator

01:28

Problem 53

Find the indicated partial sum for each sequence.
$$a_{n}=2 n-1 ; S_{8}$$

Amy Jiang

Numerade Educator

Problem 54

Find the indicated partial sum for each sequence.
$$a_{n}=3 n-1 ; S_{6}$$

Amy Jiang

Numerade Educator

00:34

Problem 55

Find the indicated partial sum for each sequence.
$$a_{n}=\frac{1}{n} ; S_{5}$$

Amy Jiang

Numerade Educator

00:51

Problem 56

Find the indicated partial sum for each sequence.
$$a_{n}=\frac{n}{n+1} ; S_{4}$$

Amy Jiang

Numerade Educator

01:13

Problem 57

Expand and evaluate each series.
$$\sum_{i=1}^{4}(3 i-5)$$

Amy Jiang

Numerade Educator

01:14

Problem 58

Expand and evaluate each series.
$$\sum_{i=1}^{5}(2 i-3)$$

Amy Jiang

Numerade Educator

01:26

Problem 59

Expand and evaluate each series.
$$\sum_{k=1}^{5}\left(2 k^{2}-3\right)$$

Amy Jiang

Numerade Educator

00:56

Problem 60

Expand and evaluate each series.
$$\sum_{k=1}^{5}\left(k^{2}+1\right)$$

Amy Jiang

Numerade Educator

00:46

Problem 61

Expand and evaluate each series.
$$\sum_{k=1}^{7}(-1)^{k} k$$

Amy Jiang

Numerade Educator

00:41

Problem 62

Expand and evaluate each series.
$$\sum_{k=1}^{5}(-1)^{k} 2^{k}$$

Amy Jiang

Numerade Educator

00:44

Problem 63

Expand and evaluate each series.
$$\sum_{i=1}^{4} \frac{i^{2}}{2}$$

Amy Jiang

Numerade Educator

00:24

Problem 64

Expand and evaluate each series.
$$\sum_{i=2}^{4} i^{2}$$

Amy Jiang

Numerade Educator

00:36

Problem 65

Expand and evaluate each series.
$$\sum_{j=3}^{7} 2 j$$

Amy Jiang

Numerade Educator

Problem 66

Expand and evaluate each series.
$$\sum_{j=3}^{7} \frac{j}{2^{j}}$$

Amy Jiang

Numerade Educator

01:53

Problem 67

Expand and evaluate each series.
$$\sum_{k=3}^{8} \frac{(-1)^{k}}{k(k-2)}$$

Amy Jiang

Numerade Educator

01:40

Problem 68

Expand and evaluate each series.
$$\sum_{k=2}^{6} \frac{(-1)^{k+1}}{k^{2}-1}$$

Amy Jiang

Numerade Educator

00:39

Problem 69

Write each sum using sigma notation. Answers are not necessarily unique.
$$4+8+12+16+20$$

Amy Jiang

Numerade Educator

00:34

Problem 70

Write each sum using sigma notation. Answers are not necessarily unique.
$$5+10+15+20+25$$

Amy Jiang

Numerade Educator

00:54

Problem 71

Write each sum using sigma notation. Answers are not necessarily unique.
$$-1+4-9+16-25+36$$

Amy Jiang

Numerade Educator

View

Problem 72

Write each sum using sigma notation. Answers are not necessarily unique.
$$1-8+27-64+125-216$$

Danielle Fairburn

Numerade Educator

00:21

Problem 73

For the given general term $a_{n},$ write the indicated sum using sigma notation.
$$a_{n}=n+3 ; S_{5}$$

Amy Jiang

Numerade Educator

00:22

Problem 74

For the given general term $a_{n},$ write the indicated sum using sigma notation.
$$a_{n}=\frac{n^{2}+1}{n+1} ; S_{4}$$

Amy Jiang

Numerade Educator

00:22

Problem 75

For the given general term $a_{n},$ write the indicated sum using sigma notation.
$a_{n}=\frac{n^{2}}{3} ;$ third partial sum

Amy Jiang

Numerade Educator

00:23

Problem 76

For the given general term $a_{n},$ write the indicated sum using sigma notation.
$a_{n}=2 n-1 ;$ sixth partial sum

Amy Jiang

Numerade Educator

00:21

Problem 77

For the given general term $a_{n},$ write the indicated sum using sigma notation.
$$a_{n}=\frac{n}{2^{n}} ; \text { sum for } n=3 \text { to } 7$$

Amy Jiang

Numerade Educator

00:16

Problem 78

For the given general term $a_{n},$ write the indicated sum using sigma notation.
$$a_{n}=n^{2} ; \text { sum for } n=2 \text { to } 6$$

Amy Jiang

Numerade Educator

01:15

Problem 79

Compute each sum by applying properties of summation.
$$\sum_{i=1}^{5}(4 i-5)$$

Amy Jiang

Numerade Educator

00:51

Problem 80

Compute each sum by applying properties of summation.
$$\sum_{i=1}^{6}(3+2 i)$$

Amy Jiang

Numerade Educator

00:52

Problem 81

Compute each sum by applying properties of summation.
$$\sum_{k=1}^{4}\left(3 k^{2}+k\right)$$

Amy Jiang

Numerade Educator

00:51

Problem 82

Compute each sum by applying properties of summation.
$$\sum_{k=1}^{4}\left(2 k^{3}+5\right)$$

Amy Jiang

Numerade Educator

01:18

$$\text { Sum of } a_{n}=3 n-2: S_{n}=\frac{n(3 n-1)}{2}$$
The sum of the first $n$ terms of the sequence defined by $a_{n}=3 n-2=1,4,7,10, \ldots$
$(3 n-2), \ldots$ is given by the formula shown. Find $S_{5}$ using the formula, then verify by direct calculation.

Amy Jiang

Numerade Educator

01:53

Problem 84

$$\text { Sum of } a_{n}=3 n-1: S_{n}=\frac{n(3 n+1)}{2}$$
The sum of the first $n$ terms of the sequence defined by $a_{n}=3 n-1=2,5,8,11, \ldots,(3 n-1), \ldots$ is
given by the formula shown. Find $S_{8}$ using the formula, then verify by direct calculation. Observing the results of Exercises 83 and $84,$ can you now state the sum formula for $a_{n}=3 n-0 ?$

Amy Jiang

Numerade Educator

Problem 85

Use the information given in each exercise to determine the $n$th term $a_{n}$ for the sequence described. Then use the $n$th term to list the specified number of terms.
Steve's car has a blue-book value of $\$ 6000 .$ Each year it loses $20 \%$ of its value (its value each year is $80 \%$ of the year before). List the value of Steve's car for the next 5 yr. (Hint: For $a_{1}=6000,$ we need the next five terms.)

Sheryl Ezze

Numerade Educator

Problem 86

Use the information given in each exercise to determine the $n$th term $a_{n}$ for the sequence described. Then use the $n$th term to list the specified number of terms.
Suppose inflation (an increase in value) will average $4 \%$ for the next 5 yr. List the growing cost (year by year) of a DVD that costs $\$ 15$ right now. (Hint: For $a_{1}=15,$ we need the next five terms.)

Sheryl Ezze

Numerade Educator

Problem 87

Use the information given in each exercise to determine the $n$th term $a_{n}$ for the sequence described. Then use the $n$th term to list the specified number of terms.
Latisha gets $\$ 5.20$ an hour for filling candy machines for Archtown Vending. Each year she receives a $\$ 0.50$ hourly raise. List Latisha's wage for the first 5 yr. How much will she make in the fifth year if she works 8 hr per day for 240 working days?

Sheryl Ezze

Numerade Educator

01:21

Problem 88

Use the information given in each exercise to determine the $n$th term $a_{n}$ for the sequence described. Then use the $n$th term to list the specified number of terms.
The average birth weight of a certain animal species is $900 \mathrm{g},$ with the baby gaining 125 g each day for the first 10 days. List the infant's weight for the first 10 days. How much does the infant weigh on the 10 th day?

Julie Silva

Numerade Educator

01:26

Problem 89

Use the information given in each exercise to determine the $n$th term $a_{n}$ for the sequence described. Then use the $n$th term to list the specified number of terms.
A local fishery stocks a large lake with 1500 bass and then adds an additional 100 mature bass per month until the lake nears maximum capacity. If the bass population grows at a rate of $5 \%$ per month through natural reproduction, the number of bass in the pond after $n$ months is given by the recursive sequence $b_{0}=1500, b_{n}=1.05 b_{n-1}+100$
How many bass will be in the lake after
6 months?

Harsh Gadhiya

Numerade Educator

01:21

Problem 90

Use the information given in each exercise to determine the $n$th term $a_{n}$ for the sequence described. Then use the $n$th term to list the specified number of terms.
The Interior Department introduces 50 wolves (male and female) into a large wildlife area in an effort to preserve the species. Each year about 12 additional adult wolves are added from capture and relocation programs. If the wolf population grows at a rate of $10 \%$ per year through natural reproduction, the number of wolves in the area after $n$ years is given by the recursive sequence $w_{0}=50, w_{n}=1.10 w_{n-1}+12 .$ How many
wolves are in the wildlife area after 6 years?

Julie Silva

Numerade Educator

00:56

Problem 91

Verify that a summation may be distributed to two (or more) sequences. That is, verify that the following statement is true:
$\sum_{i=1}^{n}\left(a_{i} \pm b_{i}\right)=\sum_{i=1}^{n} a_{i} \pm \sum_{i=1}^{n} b_{i}$

Zach Steedman

Numerade Educator

01:10

Problem 92

Surprisingly, some of the most celebrated numbers in mathematics can be represented or approximated by a series expansion. Use your calculator to find the partial sums for $n=4, n=8,$ and $n=12$ for the summations given, and attempt to name the number the summation approximates:
$$\sum_{k=0}^{n} \frac{1}{k !}$$

Amy Jiang

Numerade Educator

01:10

Problem 93

Amy Jiang

Numerade Educator

01:10

Problem 94

Amy Jiang

Numerade Educator

03:56

Problem 95

$$\text { Solve } \csc x \sin \left(\frac{\pi}{2}-x\right)=-1$$

Himanshu Kushwaha

Numerade Educator