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Unit 6 Sequences & Series

Unit 6 Learning Targets


Number Chapter and Section Title Description

LT1 10.1 Sequences and Summation Notation

Find particular terms of a general or recursive sequence using the general formula.


Evaluate a factorial expression.


Given terms in a sequence, create the general or recursive formula.


Given summation notation, evaluate the sum.


Express a sum using summation notation.


Given a set of terms, write the sum using sigma notation.

LT7 10.2 Arithmetic Sequences Find particular terms of an arithmetic sequence using the general formula.

LT8 10.2 Arithmetic Sequences Determine the common difference for an arithmetic sequence.

LT9 10.2 Arithmetic Sequences Given terms in a sequence, create the arithmetic formula.

LT10 10.2 Arithmetic Sequences Evaluate the sum of a finite arithmetic series.

LT11 10.3 Geometric Sequences Find particular terms of a geometric sequence using the general formula.

LT12 10.3 Geometric Sequences Determine the common ratio for a geometric sequence.

LT13 10.3 Geometric Sequences Given terms in a sequence, create the geometric formula.

LT14 10.3 Geometric Sequences Evaluate the sum of a finite or infinite geometric series.

LT15 10.3 Geometric Sequences Identify if a sequence is arithmetic, geometric, or neither.

LT16 10.5 Binomial Theorem Evaluate a binomial coefficient.


LT17 10.5 Binomial Theorem Expand a binomial raised to a power.

LT18 10.5 Binomial Theorem Find a particular term in a binomial expansion.

LT19 10.5 Binomial Theorem Find the coefficient only of the indicated term of the given binomial.

Learning Targets 1-6 Unit 6: Sequences and Summation Notation Practice 10.1

Unit 6 Practice Page 1

Write the first four terms of each sequence whose general term is given.

1. an = 3n + 2 2. an = 3𝑛

_____ _____ _____ _____ _____ _____ _____ _____

3. an = (−3)𝑛 4. an = (−1)𝑛(𝑛 + 3)

_____ _____ _____ _____ _____ _____ _____ _____

5. an =2𝑛

𝑛+4 6. an =

(−1)𝑛+1

2𝑛−1

_____ _____ _____ _____ _____ _____ _____ _____

Write the first four terms of each sequence defined using recursion formulas.

7. an = 𝑎𝑛−1 + 5 𝑓𝑜𝑟 𝑛 ≥ 2 𝑖𝑓 𝑎1 = 7 8. an = 4𝑎𝑛−1 𝑓𝑜𝑟 𝑛 ≥ 2 𝑖𝑓 𝑎1 = 3

_____ _____ _____ _____ _____ _____ _____ _____

9. an = 2𝑎𝑛−1 + 3 𝑓𝑜𝑟 𝑛 ≥ 2 𝑖𝑓 𝑎1 = 4 10. an = 3𝑎𝑛−1 − 1 𝑓𝑜𝑟 𝑛 ≥ 2 𝑖𝑓 𝑎1 = 5

_____ _____ _____ _____ _____ _____ _____ _____



The general term of a sequence is given and involves a factorial. Write the first four terms of each.

11. an =𝑛2

𝑛! 12. an = 2(n + 1)!

_____ _____ _____ _____ _____ _____ _____ _____

Evaluate each factorial expression. Show your work.

13. 17!

15! __________ 14.

16!

2!14! __________

15. (n+2)!

n! __________ 16.

(2n+1)!

(2n)! __________

17. A deposit of $6000 is made in an account that earns 6% interest compounded quarterly. The balance in the

account after n quarters is given by the sequence

an = 6000 (1 +0.06

4)

𝑛

, 𝑛 = 1, 2, 3, …

Find the balance in the account after 5 years. (Hint: How many quarters are in 5 years?)



Expand and evaluate each sum.

18. ∑ 5𝑖 =

6

𝑖=1

19. ∑ 2𝑖2 =

4

𝑖=1

20. ∑ 𝑘(𝑘 + 4) =

5

𝑘=1

21. ∑ (−1

2)

𝑖

=

4

𝑖=1

22. ∑ 11 =

9

𝑖=5

23. ∑(−1)𝑖

𝑖!=

4

𝑖=0

24. ∑𝑖!

(𝑖 − 1)!=

5

𝑖=1

25. ∑(−1)𝑖+1

(𝑖 + 1)!=

4

𝑖=0



Express each sum using summation notation. Use 1 as the lower limit and i for the index of summation.

26. 12 + 22 + 32 + ⋯ + 152 27. 2 + 22 + 23 + ⋯ + 211

28. 1 + 2 + 3 + ⋯ + 30 29. 1

2+

2

3+

3

4+ ⋯ +

14

14+1

30. 4 + 42

2+

43

3+ ⋯ +

4𝑛

𝑛 31. 1 + 3 + 5 … + (2𝑛 − 1)

Express each sum using summation notation. Use a lower limit of summation and index of summation of your

choice.

32. 5 + 7 + 9 + 11 … + 31 33. 𝑎 + 𝑎𝑟 + 𝑎𝑟2 + ⋯ + 𝑎𝑟12

34. 6 + 9 + 12 + 15 … + 33

35. (𝑎 + 𝑑) + (𝑎 + 𝑑2) + (𝑎 + 𝑑3) + (𝑎 + 𝑑4) + ⋯ + (𝑎 + 𝑑𝑛)

Learning Targets 7-10 Unit 6: Arithmetic Sequences Practice 10.2


Write the first five terms of each arithmetic sequence.

1. 𝑎1 = 300 and 𝑑 = −90 2. 𝑎1 =5

2 and 𝑑 = −

1

2

_____ _____ _____ _____ _____ _____ _____ _____ _____ _____

3. 𝑎𝑛 = 𝑎𝑛−1 + 6 if 𝑎1 = −9 4. 𝑎𝑛 = 𝑎𝑛−1 +1

2 if 𝑎1 = −1

_____ _____ _____ _____ _____ _____ _____ _____ _____ _____

Find the indicated term of the arithmetic sequence.

5. 𝑎1 = 13 and 𝑑 = 4 𝑎6 = ________ 6. 𝑎1 = 7 and 𝑑 = 5 𝑎50 = ________

7. 𝑎5 = −40 and 𝑎15 = −50 𝑎200 = ________ 8. 𝑎20 = 35 and 𝑑 = −3 𝑎60 = ________

9. 𝑎16 = −60 and 𝑎40 = −48 𝑎150 = ________ 10. 𝑎12 = −32 and 𝑑 = 4 𝑎70 = _______

11. 𝑎5 = 12 and 𝑎50 = 147 𝑎92 = __________ 12. 𝑎7 = 10 and 𝑑 = 0.5 𝑎70 = _______



Write a formula for the general term of each arithmetic sequence. Do not use a recursive formula. Then find the

20th

term of the sequence.

13. 1, 5, 9, 13, … 14. 7, 3, -1, -5, …

𝑎𝑛 = ___________________ 𝑎20 = __________ 𝑎𝑛 = ___________________ 𝑎20 = ________

15. 𝑎5 = 9 and 𝑑 = 2 16. 𝑎12 = −20 and 𝑑 = −4

𝑎𝑛 = ___________________ 𝑎20 = __________ 𝑎𝑛 = ___________________ 𝑎20 = ________

17. 𝑎𝑛 = 𝑎𝑛−1 + 3 if 𝑎1 = 4 18. 𝑎𝑛 = 𝑎𝑛−1 − 10 if 𝑎1 = 30

𝑎𝑛 = ___________________ 𝑎20 = __________ 𝑎𝑛 = ___________________ 𝑎20 = ________

19. In 1970, the median age of first marriage for U.S. men was 23.2. On average, this age has increased by

approximately 0.12 per year.

a. Write a formula for the nth term of the arithmetic sequence that describes the median age of first marriage for

U.S. men n years after 1969.

b. What will be the median age of the first marriage for U.S. men in 2009?



Find the following sums.

20. The first 20 terms of the arithmetic sequence: 4, 10, 16, 22, . . . Sum = __________

21. The first 50 terms of the arithmetic sequence: –10, –6, –2, 2 . . . Sum = __________

22. The first 100 natural numbers. (Hint: 1, 2, 3, 4, . . ., 100) Sum = __________

23. The first 60 positive even integers. Sum = __________

24. The even integers between 21 and 45. Sum = __________

25. ∑(5𝑖 + 3) =

17

𝑖=1

26. ∑(−3𝑖 + 5) =

30

𝑖=1

Sum = __________ Sum = __________

27. ∑ 4𝑖 =

100

𝑖=1

28. ∑(1

2𝑖 − 5) =

42

𝑖=0

Sum = __________ Sum = __________



29. Mrs. Biberdorf wants to know how many people can sit in her church’s pews.

The first row holds 4 people, the second row holds 7 people, the third row holds 10 people and so on. Her

church has 25 rows of pews.

How many people can sit in her church’s pews?

30. Mrs. Neal is organizing cupcakes into the shape of a Christmas tree. She wants to top row to have one

cupcake, the second row to have two cupcakes, the third row to have three cupcakes, and so on. She wants the

final Christmas tree to have a total of 12 rows. How many cupcakes must she bake?

Learning Targets 1-10 Unit 6: Arithmetic Sequences Review Ch. 10.1 & 10.2 Set 1


Write the first four terms of the sequence

1. 𝑎𝑛 = (−4)𝑛 2. 𝑎𝑛 =𝑛!

𝑛3

_____ _____ _____ _____ _____ _____ _____ _____

3. 𝑎𝑛 = 4𝑎𝑛−1 − 𝑛 𝑖𝑓 𝑎1 = 1

_____ _____ _____ _____

Simplify the ratio of factorials. Show your work.

4. 32!

29! ________________ 5.

𝑘!

(𝑘−2)! ________________

Find the general formula for each sequence.

6. 1, 5, 9, 13, 17, . . . 𝑎𝑛 = _________________

7. 3,3

8,

3

27 ,

3

64 ,

3

125 , … 𝑎𝑛 = _________________

8. 2, –4, 6, –8, 10, . . . 𝑎𝑛 = _________________

9. 2

7,

5

7 ,

10

7 ,

17

7 ,

26

7, … 𝑎𝑛 = _________________

10. 𝑎1 = 5 𝑎𝑛𝑑 𝑑 = 3 𝑎𝑛 = _________________

11. 𝑎4 = 9 𝑎𝑛𝑑 𝑎10 = −15 𝑎𝑛 = _________________



Find the sum.

12. ∑(𝑘2 − 2) = _________________

7

𝑘=3

13. ∑ 8 =

3

𝑘=1

_________________

14. ∑ (1

𝑗!) = _________________

4

𝑗=2

15. ∑(−2𝑖 + 6) =

25

𝑖=1

_________________

16. The first 22 terms of the arithmetic sequence 5, 12, 19, 26… _______________

Use summation notation (sigma notation) to write the following sums.

17. 1 + 3 + 5 + 7 + … + [2(12) - 1] _______________

18. [3 + (1

3)

2] + [3 + (

1

4)

2] + [3 + (

1

5)

2] + ⋯ + [3 + (

1

9)

2] _______________

19. 1 – 2 + 4 – 8 + 16 – 32 + 64 – 128 _______________

20. 3

2+

9

4+

27

8+

81

16+

243

32 _______________

21. 3 + 8 + 15 + 24 + 35 _______________



Write the first five terms of the sequence

1. 𝑎𝑛 = 𝑛2 + 1 2. 𝑎𝑛 =𝑛2

𝑛!

_____ _____ _____ _____ _____ _____ _____ _____ _____ _____

3. 𝑎𝑛 = 3𝑎𝑛−1 − 𝑛 𝑖𝑓 𝑎1 = 1 4. 𝑎𝑛 = 𝑛! − 1

_____ _____ _____ _____ _____ _____ _____ _____ _____ _____


5. 43!

47! __________ 6.

(𝑛+1)!

(𝑛−1)! __________


7. 𝑑 =1

3 𝑎𝑛𝑑 𝑎1 =

1

3 𝑎𝑛 = _________________

8. 0,7

3,

26

3 ,

63

3 , … 𝑎𝑛 = _________________

9. –5, 10, –15, 20, . . . 𝑎𝑛 = _________________

10. 𝑑 = −9 𝑎𝑛𝑑 𝑎1 = 33 𝑎𝑛 = _________________

11. 𝑎5 = 129 𝑎𝑛𝑑 𝑎17 = −15 𝑎𝑛 = _________________



Find the sum.

12. ∑(7𝑘 + 5) = _________________

10

𝑘=0

13. ∑ 5 =

21

𝑘=1

_________________

14. ∑(𝑛! + 4) = _________________

5

𝑗=1

15. ∑ 𝑖 =

18

𝑖=1

_________________

16. The first 25 terms of the arithmetic sequence 3, 6, 9, 12, … _______________


17. 15 + 18 + 21 + 24 + … + 54 _______________

18. [5 +1

2] + [5 +

2

9] + [5 +

2

16] + ⋯ + [5 +

2

81] _______________

19. – 2 + 6 – 10 + 14 – 18 _______________

20. Find the seating capacity of an auditorium with 30 rows of seats if there are 20 seats in the first row, 27 seats in

the second row, 34 seats in the third row and so on.

_______________



Write the first five terms of the sequence

1. 𝑎𝑛 =(−1)𝑛

2𝑛2 2. 𝑎𝑛 =𝑛!

3𝑛−1

_____ _____ _____ _____ _____ _____ _____ _____ _____ _____

3. 𝑎𝑛 = 𝑎𝑛−1(𝑛 + 2) 𝑖𝑓 𝑎1 = 4

_____ _____ _____ _____ _____


4. 12!

9! ____________ 5.

(𝑛+3)!

(𝑛+1)! ____________


6. 2, 4, 8, 16, 32, . . . 𝑎𝑛 = _________________

7. −1

3,

1

9 , −

1

27 ,

1

81 , … 𝑎𝑛 = _________________

8. 10, 3, –4, –11, . . . 𝑎𝑛 = _________________

9. 𝑎𝑛 = 𝑎𝑛−1 + 3 𝑖𝑓 𝑎1 = 4 𝑎𝑛 = _________________

Find the sum.

10. ∑1

𝑘2 = _________________

4

𝑘=1

11. ∑2𝑘

𝑘 + 1=

3

𝑘=0

_________________



12. ∑(3𝑖 + 4) = _________________

30

𝑘=1

13. ∑𝑘 + 5

3=

14

𝑘=0

_________________

14. The first 40 terms of the arithmetic sequence 7, 10, 13, 16… _______________

15. The first 40 terms of the arithmetic sequence with 𝑎1 = 10 and 𝑎12 = 32. _______________


16. 1

4−

1

8+

1

16−

1

32+

1

64 _______________

17. 1

1+

1

4+

2

16+

6

64+

24

256 _______________

Using the given information for an arithmetic sequence, find the missing value.

18. 𝑎6 = 21 𝑎𝑛𝑑 𝑑 = 4 19. 𝑎4 = 8 𝑎𝑛𝑑 𝑎12 = 12

𝑎20 = ____________ 𝑎22 = ____________

20. 𝑎5 = 10 𝑎𝑛𝑑 𝑑 = 3 21. 𝑎5 = 8.5 𝑎𝑛𝑑 𝑎12 = 19

𝑎25 = ____________ 𝑎9 = ____________

Learning Targets 11-15 Unit 6: Geometric Sequences Practice 10.3


Write the first five terms of each geometric sequence.

1. 𝑎1 = 5 𝑎𝑛𝑑 𝑟 = 3 2. 𝑎1 = 20 𝑎𝑛𝑑 𝑟 =1

2

_____ _____ _____ _____ _____ _____ _____ _____ _____ _____

3. 𝑎𝑛 = −4𝑎𝑛−1 𝑖𝑓 𝑎1 = 10 4. 𝑎𝑛 = −5𝑎𝑛−1 𝑖𝑓 𝑎1 = −6

_____ _____ _____ _____ _____ _____ _____ _____ _____ _____

5. 𝑎𝑛 = −3𝑎𝑛−1 𝑖𝑓 𝑎1 = 10

_____ _____ _____ _____ _____

Use the formula for the general term of a geometric sequence to find the indicated term.

6. 𝑎1 = 6 𝑎𝑛𝑑 𝑟 = 2 7. 𝑎1 =1

4 𝑎𝑛𝑑 𝑎5 = 4

𝑎8 = ____________ 𝑎12 = ____________

8. 𝑎2 = 10 𝑎𝑛𝑑 𝑎5 = 0.01 9. 𝑎3 =2

3 𝑎𝑛𝑑 𝑎5 =

2

27

𝑎8 = ____________ 𝑎8 = ____________

10. 𝑎1 = 4 𝑎𝑛𝑑 𝑟 = −2 11. 3, 15, 75, 375, …

𝑎12 = ____________ 𝑎7 = ___________



Write a general formula for each geometric sequence. Use the formula to find 𝑎7 (the 7th

term).

12. 3, 12, 48, 192, … 13. 18, 6, 2,2

3, …

𝑎𝑛 = _____________________ 𝑎7 = __________ 𝑎𝑛 = _____________________ 𝑎7 = __________

14. 1.5, −3, 6, −12, … 15. 0.0004, −0.004, 0.04, 0.04, −0.4 …

𝑎𝑛 = _____________________ 𝑎7 = __________ 𝑎𝑛 = _____________________ 𝑎7 = __________

16. Suppose you save $1 on the first day of a month, $2 the second day, $4 the third day, and so on. That is, each

day you save twice as much as you did the day before. What will you put aside on the fifteen day of the month?

17. A professional baseball player signs a contract with a beginning salary of $3,000,000 for the first year and an

annual increase of 4% per year beginning in the second year. That is, beginning in year 2, the athlete’s salary will

be 1.04 times what it was in the previous year. What is the athlete’s salary for year 7 of the contract?



Find the sums.

18. First 12 terms of 2, 6, 18, 54, . . . 19. First 11 terms of 3, –6, 12, –24, . . .

Sum = __________ Sum = __________

20. First 14 terms of −3

2, 3, −6, 12 … 21. ∑(3)𝑖 =

8

𝑖=1

Sum = __________ Sum = __________

22. ∑ 5 ∙ 2𝑖 =

10

𝑖=1

23. ∑ (1

2)

𝑖+1

=

6

𝑖=1

Sum = __________ Sum = __________

Find the sum of each infinite geometric series.

24. 1 + 1

3+

1

9+

1

27+ ⋯ 25. 3 +

3

4+

3

42 +3

43 + ⋯

Sum = __________ Sum = __________



26. 1 −1

2+

1

4, −

1

8+ ⋯ 27. ∑ 8(−0.3)𝑖−1

∞

𝑖=1

Sum = __________ Sum = __________

The general term of a sequence is given. Determine whether the sequence is arithmetic, geometric or neither. If it

is arithmetic, state the common difference. If it is geometric, state the common ratio.

28. 𝑎𝑛 = 𝑛 + 5 29. 𝑎𝑛 = 2𝑛 30. 𝑎𝑛 = 𝑛2 + 5

Use the formula for the value of an annuity to solve. Hint: 𝐴 =𝑃[(1+

𝑟

𝑛)

𝑛𝑡−1]

𝑟

𝑛

P is the deposit made at the end of each compounding period

r is the percent interest compounded n times per year

A is the value of the annuity after t years

31. To save for retirement, you decide to deposit $2500 into an IRA at the end of each year for the next 40 years.

If the interest rate is 9% per year compounded annually, find the value of the IRA after 40 years.

32. You decide to deposit $100 at the end of each month into an account paying 8% interest compounded

monthly to save for your child’s education. How much will you save over 16 years?

Learning Targets 1-15 Unit 6: Sequences & Series Review 10.1-10.3 Set 1


Find the first five terms of each sequence.

1. 𝑎𝑛 = (−1)𝑛 (𝑛

𝑛+2) 2. 𝑎𝑘 = 𝑎𝑘−1 + 3𝑘 𝑖𝑓 𝑎1 = 6

_____ _____ _____ _____ _____ _____ _____ _____ _____ _____

Write an expression for the most apparent nth term of the sequence.

3. 1

3, −

1

6,

1

9, −

1

12,

1

15, … 𝑎𝑛=

Use sigma notation to write the given sum.

4. 1

4+

2

8+

3

16+ ⋯ +

6

128 _________________________

Using the given information for an ARITHMETIC sequence to find the missing value.

5. 𝑎4 = −10 𝑎𝑛𝑑 𝑑 = 4 𝑎12 = ________ 6. 𝑎6 = 18 𝑎𝑛𝑑 𝑎10 = −6 𝑎15 =________

7. Find the sum of the first 15 terms of the arithmetic sequence if 𝑎1 = 4 𝑎𝑛𝑑 𝑎15 = 88.

S = __________

Use the given information for a GEOMETRIC sequence to find the missing value.

8. 𝑎1 = 2 𝑎𝑛𝑑 𝑟 = 1.1 𝑎8 = _______ 9. 𝑎4 = 3 𝑎𝑛𝑑 𝑎9 =1024

81 𝑎1 = ______



Find the sum. (HINT: First determine if the sequence is arithmetic, geometric or infinite geometric.)

10. First 20 terms of 8 + 11 + 14 + 17 + … 11. First 15 terms of 2 + 6 + 18 + 54 + …

S = ____________ S = ____________

12. ∑ 3 (3

2)

𝑛

=

500

𝑛=1

13. ∑(2𝑛 − 1) =

100

𝑛=0

S = ____________ S = ____________

14. ∑4

𝑘!=

3

𝑘=0

15. ∑𝑘2

3𝑘 + 2=

6

𝑘=2

S = ____________ S = ____________

16. ∑ 2 (1

2)

𝑛

=

∞

𝑛=1

17. 4 − 1 +1

4−

1

16+ ⋯

S = ____________ S = ____________



Write the first five terms of each sequence.

1. 𝑎𝑘 = −1

4𝑎𝑘−1 𝑖𝑓 𝑎1 = 4 2. 𝑎𝑛 =

2𝑛

𝑛2

______ ______ ______ ______ ______ ______ ______ ______ ______ ______

Simplify.

3. (𝑛+2)!

(𝑛−1)! _______________ 4.

82!

85! _______________

Write an expression for the most apparent nth term of these sequences.

5. 3

2,

5

4,

7

6,

9

8,

11

10, … 𝑎𝑛=

6. 1, 2, 4, 8, … 𝑎𝑛=

7. 1, −1, −3, −5, … 𝑎𝑛=

Find the sum of these sequences.

8. The first 18 terms of 1, –3, –7, –11, … 9. The first 8 terms of 3 + 1 +1

3+

1

9+ ⋯

S = ______________ S = ______________

10. ∑ 4 (1

2)

𝑘−1∞

𝑘=1

S = ______________

Learning Targets 16-19 Unit 6: Binomial Theorem Practice 10.5


Evaluate each binomial coefficient.

1. (83) = _____ 2. (12

1) = _____ 3. (6

6) = _____ 4. (100

2) = _____

Use the Binomial Theorem to expand each binomial and express in simplified form.

5. (𝑥 + 2)3 = ________________________________________

6. (5𝑥 − 1)3 = ________________________________________

7. (𝑥2 + 2𝑦)4 = ________________________________________

8. (2𝑥3 − 1)4 = ________________________________________

Learning Targets 16-19 Unit 6: Binomial Theorem Practice 10.5


Use the Binomial Theorem to expand each binomial. Express the result in simplified form.

9. (3𝑥 + 𝑦)3 10. (𝑦 − 3)4

__________________________________ __________________________________

Write the first three terms in each binomial expansion. Express the result in simplified form.

11. (𝑥 + 2)8 12. (𝑥2 + 1)16

__________________________________ __________________________________

Find the term indicated in each expansion.

13. (2𝑥 + 𝑦)6 Third term__________ 14. (𝑥2 − 𝑦3)8 Sixth term__________

15. (𝑥2 + 𝑦)22 𝑦14 term __________ 16. (𝑥2 − 2𝑦)12 𝑥8 term __________

Learning Targets 16-19 Unit 6: Binomial Theorem More Practice 10.5


Expand each binomial. Simplify.

1. (3𝑥 + 2)3 _______________________________________________________

2. (2𝑥 + 7)5 _______________________________________________________

3. (4𝑥 − 𝑦)4 _______________________________________________________

4. Find the coefficient of 𝑥3 in the expansion of (𝑥 + 2)8. _________________

5. Find the coefficient of 𝑥6 in the expansion of (2𝑥 − 3)14. _________________

6. Find the coefficient of 𝑥3𝑦6 in the expansion of (𝑥 − 6𝑦)9. _________________

7. Find the coefficient of 𝑥5𝑦7 in the expansion of (3𝑥 + 4)12. _________________

Learning Targets 1-19 Unit 6: Sequences and Series Practice Test


Write the first four terms of each sequence.

1. 𝑎𝑛 = (−3)𝑛 __________ __________ __________ __________

2. 𝑎𝑘 = 2𝑎𝑘−1 − 5 𝑖𝑓 𝑎1 = 2 __________ __________ __________ __________

Simplify each ratio of factorials. Leave answers as fractions (not decimals) if possible.

3. 18!

20! _______________ 4.

(3𝑛+1)!

(3𝑛−1)! _______________

Write an expression for the most apparent nth term.

5. 3, −6, 9, −12, 15 6. 4, 9, 14, 19, 24

𝑎𝑛= 𝑎𝑛=

Find each sum.

7. ∑ 3 = _________________

5

𝑘=1

8. ∑(3 − 4𝑘) =

3

𝑘=0

_________________

Use sigma notation to write the sum.

9. [2 + (1

2)

2] + [2 + (

2

3)

2] + [2 + (

3

4)

2] + ⋯ + [2 + (

9

10)

2] ______________________

10. 2

8+

2

27+

2

64+ ⋯ +

2

343 ______________________



Using the given information for an arithmetic sequence, find the missing value.

11. 𝑎1 = 4 𝑎𝑛𝑑 𝑑 =1

3 12. 𝑎3 = 1.25 𝑎𝑛𝑑 𝑎7 = −3.15

𝑎13 = _________________ 𝑎10 = _________________

13. Find the sum of the first 15 terms of the arithmetic sequence given 𝑎1 = 4 𝑎𝑛𝑑 𝑎15 = 88.

S = _________________

Using the given information for a geometric sequence, find the missing value.

14. 𝑎1 = 2 𝑎𝑛𝑑 𝑟 = 1.1 15. 𝑎4 = 3 𝑎𝑛𝑑 𝑎9 =1024

81

𝑎8 = _________________ 𝑎1 = _________________

Find each sum. (Hint – determine whether the sequence is arithmetic or geometric, first)

16. The first 20 terms of 8 + 11 + 14 + 17 + ⋯ 17. ∑ 3 (1

3)

𝑛

=

40

𝑛=1

S = _________________ S = _________________

18. The first 15 terms of 2 + 6 + 18 + 54 + ⋯ 19. ∑(6𝑛 − 4) =

250

𝑛=0

S = _________________ S = _________________



20. Find the sum of the infinite geometric series: 4 − 1 +1

4−

1

16+ ⋯ S = ________________

21. Determine the seating capacity of an auditorium with 36 rows of seats if there are 16 seats in the first row, 18

seats in the second row, 20 seats in the third row, and so on.

S = _________________

22. Evaluate: (4038

) = ___________

Expand each binomial. Simplify.

23. (𝑥 + 2𝑦)5 _______________________________________________________

24. (2 − 3𝑖)4 _______________________________________________________

25. Find the 4th

term in the expansion of (𝑥

2+ 3)

8. ____________________



Find the coefficient of the given term in the expansion of each binomial.

26. the 𝑥5𝑦7 term of (𝑥 + 5𝑦)12 ____________________

27. the 𝑥4 term of (3𝑥 − 2)11 ___________________

28. the 𝑥8 term of (𝑥2 + 4)6 ____________________

Learning Targets 1-19 Unit 6: Sequences and Series Extra Review


1. Find the 1st

term of the arithmetic sequence with 𝑎16 = 22 𝑎𝑛𝑑 𝑎10 = 72. __________________

2. Find the 10th

term of the geometric sequence with 𝑎2 = 3 𝑎𝑛𝑑 𝑎5 =3

64. __________________

3. Find the 30th

term of this sequence: 5, 2, –1, –4, . . . __________________

4. Find the 20th

term of this sequence: 4, 10, 25, 62.5, . . . __________________

Find each sum.

5. ∑ 4 (4

3)

𝑘

= _________________

13

𝑘=0

6. ∑3𝑘 + 1

4=

25

𝑘=1

_______________ 7. ∑(2𝑘2 − 1) =

4

𝑘=0

___________

8. Find the sum of the infinite geometric series: 3 − 1 +1

3−

1

9+ ⋯ S = _____________

Learning Targets 1-19 Unit 6: Sequences and Series Extra Review


Find the coefficient of the given term in the expansion of each binomial.

9. the 𝑥2𝑦8 term of (4𝑥 − 𝑦)10 ____________________

10. the 𝑥10 term of (𝑥2 − 3)8 _____________________

11. Find the 8th

term in the expansion of (2𝑥 − 3)14. _____________________

12. Find the 6th

term in the expansion of (4𝑥 + 1)10. _____________________

13. Expand and simplify: (𝑥 − 2𝑦)5 ___________________________________________________

Write a formula for the most apparent nth term. Remember, n must start with 1.

14. 1

1, −

8

2,

27

3, −

64

4, … 15. 4, 9, 16, 25, . . .

____________________ ____________________

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