HONORS A2-TRIG SEQUENCES AND SERIES UNIT 6

SEQUENCES (DAY 1)

Warm-Up: Solve for x to the nearest hundredth: 5

4x

1x

1

Sequence: A set of ________________ written in a given ______________.

Notation: 1a , 2a , 3a , 4a , …, 1na , na , 1na

Types of Sequences:

Arithmetic Sequence: a set of _________________ in a specific order each

having a __________________ difference.

o Exampl e: 1, 6, 11, 16, 21,… (common difference = ______)

Geometric Sequence: a set of ________________ in a specific order having

a common _____________.

o Example: 1, 2, 4, 8, 16,… (common ratio = _______)

Ways to Express Sequences:

Explicit Formula: A formula that allows computation of _____ term (an).

o Example: an = 2n + 5

n an = 2n + 5 an

1

2

3

4

5

n

n = term

number

1a = first term

na = the term

at the

number (n)

2

Recursive Formula: A formula that allows any term (an) to be computed

from the _______________ term (an -1). You must be given the ___________

term (a1).

o Example: a1 = 5

an = an-1 + 4

1) Write the first three terms of the sequence an = 3n – 1.

2) List the next three terms of the sequence: a1 = 5

an = 2an-1 + 3

n an = an-1 + 4 an

1

2

3

4

5

n

n = term

number

1a = first term

na = the term

at the

number (n)

1na = the term

before na

3

In 3-6, a sequence is described.

a) Identify the definition as recursive or explicit.

b) Find the first three terms of the sequence.

3) r1 = -4 4) tn = -6n3 + 27n2 – 48n + 23

rn = -3rn-1

5) j1 = 3 6) kn = 12 – 3(n – 1)

jn+1 = (jn)2

7) Write both a recursive and an explicit definition for the following sequence:

4, 8, 12, 16, 20

8) Given the sequence: 2, 4, 8, 16,…

a) List the next three terms of the sequence.

b) Write an explicit formula for an.

c) Write a recursive definition for the sequence.

9) Erin is memorizing words for a vocabulary test. She currently knows 25 words. Each

day for the next 6 days, she increased the words memorized by 3.

a) Write the sequence for the number of words that Erin memorized for each of

the seven days.

b) Write a recursive definition for this sequence.

4

ARITHMETIC SEQUENCES (DAY 2)

Warm-Up: Find the center and radius of the circle whose equation is:

030y10x6yx 22

Arithmetic Sequence: a set of _________________ in a specific order each having a

__________________ difference.

Example: 1, 3, 5, 7, 9,… (common difference = ______)

Determine if each sequence is arithmetic, and if so, find the common difference.

1) 5, 11, 17, 23, 29,… 2) 4, 8, 16, 32, 64,…

3) 1, 4, 9, 16, 25, 36,… 4) 20, 16, 12, 8, 4,…

5) For the sequence 100, 97, 94, 91, …, find:

a) the common difference.

b) the 20th term of the sequence.

To find any missing information for an arithmetic sequence, use the formula:

an = a1 + (n – 1)d

5

6) Cooper is saving to buy a guitar. In the first week, he put aside $42 that he received

for his birthday, and in each of the following weeks, he added $8 to his savings. He

needs $400 for the guitar that he wants. In which week will he have enough money

for the guitar?

7) Given the arithmetic sequence, 6, 10, 14, 18, 22, … , find the 94th term.

8) The 4th term of an arithmetic sequence is 80 and the 12th term is 32.

a) What is the common difference?

b) What is the first term of the sequence?

9) Write an explicit formula of the arithmetic sequence that has two given terms

of a4 = 36 and a9 = 61.

6

ARITHMETIC MEANS

Arithmetic Mean: Any term between two other terms in an arithmetic sequence.

The arithmetic mean is the average of the term ___________ and

_____________ it.

Example: Using the sequence: 2, 5, 8, 11, 14,…

8 is the arithmetic mean between _____ and _____.

When given two non-consecutive terms, and asked to find the arithmetic means:

1) Find the common difference by using the formula:

term#term#

termtermd

2) Find the remaining terms.

1) Given the arithmetic sequence: 4, a2, a3, a4, 28.

a) Find the common difference.

b) Find the three missing arithmetic means.

2) Find five arithmetic means between 2 and 23.

3) Hannah noticed that the daily number of text messages she received over the

course of two months form an arithmetic sequence. If she received 13 messages on

day 3 and 64 messages on day 20, find how many messages Hannah received on,

a) day 12? b) day 50?

7

ARITHMETIC SERIES (DAY 3)

Warm-Up: Simplify: 135616 ii2ii

Arithmetic Series: The ___________ of the terms in an ______________________ sequence.

Example: Find the sum of the sequence: 1, 2, 3, 4, 5, …, 99, 100

1) A student borrowed $4,000 for college expenses. The loan is to be repaid over a 100-

month period with monthly payments of $60.00, $59.80, $59.60, …, $40.20. How much

did the student pay over the life of the loan?

2) An auditorium has 21 rows of seats. The first row has 18 seats, and each succeeding

row has two more seats than the previous row. How many seats are in the

auditorium?

3) The sum of the first and the last terms of an arithmetic sequence is 80 and the sum of

all the terms is 1,200. How many terms are in the sequence?

Arithmetic Series Formulas

To find the sum of any arithmetic sequence, use the formula:

Sn = 2

)n

a1

a(n

8

4) Find the sum of the first ten terms of the series: 5 + –2 + –9 + –16 +…

5) Jack Deere made a deal with this father that he would mow the lawn for the entire

summer. For the first mowing, he would charge a special introductory rate of $2, but

for each mowing thereafter he would charge 50 cents more than the previous rate. If

Jack mowed the lawn a total of 48 times during the summer, how much did he earn

in all?

6) To build up endurance, Arnold started an exercise program in which he exercised 30

minutes the first day, 34 minutes the next day, then 38 minutes, 42 minutes, and so on,

each day extending his exercise time by 4 minutes. If he continued at this rate,

ending at 2 hours, 30 minutes, what was the total time he spent exercising?

7) Find the sum of the first 100 positive odd integers

8) Sophia wanted to start putting some money away for a rainy day. She bought two

piggy banks to help her out. In the first piggy bank she put away two dollars per day

for 365 days. In the second piggy bank she put away a quarter on the first day and

increased her daily deposit by a dime every day after that for 365 days.

a) How much money will Sophia have in her first piggy bank after 365 days?

b) How much money will she have in her second piggy bank after 365 days?

9

GEOMETRIC SEQUENCES (DAY 4)

Warm-Up: Find the exact roots of the equation 0128x64x2x 23

Geometric Sequence: a set of ________________ in a specific order having a

common _____________ between all terms.

Example: 1, 2, 4, 8, 16,… (common ratio = _______)

1) Given the sequence: 4, 12, 36, 108, 324, …

a) Is this a geometric sequence?

b) What is the 10th term of this sequence?

In 2 – 5, a sequence is described.

a) Determine if they are geometric, arithmetic, or neither.

b) If the sequence is geometric, identify the common ratio.

2) 5, 10, 15, 20, 25, … 3) 10, 30, 90, 270, 810, …

4) 64, 32, 16, 8, 4, … 5) 1, 4, 9, 16, 25, …

6) 5x, 10x, 15x, 20x, … 7) ,...x,x,x,x 11975

To find any missing information for a geometric

sequence, use the formula:

an = (a1)(r)n-1

10

8) What is the common ratio of the sequence ,...ab16

9,ba

32

3,ba

64

1 54335 ?

(1) 2a2

b3 (3)

b

a3 2

(2) 2a

b6 (4)

b

a6 2

9) Given the geometric sequence 1, 5, 25, 125, …

a) Write an explicit formula for finding the nth term of the sequence.

b) Use this formula, to find the 8th term of the sequence.

10) Write an explicit formula of the geometric sequence that has two given terms

of a4 = 4 and a7 = 32.

11

Geometric Means

Geometric Mean: In a geometric sequence, any term between two other terms

(the term _________ it and the term ________ it) is the geometric

____________ of those two terms.

o Example: Using the sequence: 3, 6, 12, 24, 48,…

12 is the geometric mean between _____ and _____.

That means that: x

x

When given two non-consecutive terms, and asked to find the geometric means:

1) Find the common ratio by using the formula:

term

term

term#r

term#r

2) Find the remaining terms.

1) Find four geometric means between 5 and 1,215.

2) Form a sequence that has three geometric means between 4 and 324.

3) A sequence has the following terms: 5.62a,25a,10a,4a 4321

Which formula represents the nth term in this sequence?

(1) n5.24an (3) nn )5.2(4a

(2) )1n(5.24an (4) 1nn )5.2(4a

12

GEOMETRIC SERIES (DAY 5)

Warm-Up: Simplify:

22 n

1

m

1n

1

m

1

Geometric Series: The ___________ of the terms in a ______________________ sequence.

Example: Find the sum of the sequence: 3, 12, 48,…, 768, 3072

1) Find the sum of the first eight terms of the series 3 – 12 + 48 – 192 + …

2) Find the sum of eight terms of the geometric series whose first term is 2 and

whose eighth term is 4,374.

3) Find the sum of the first six terms of the geometric series whose first term is 13 and

whose sixth term is 13,312.

Geometric Series Formula

To find the sum of any geometric sequence, use the formula:

Sn = r1

)nr1(1

a

13

4) Find the sum of five terms of the geometric series whose first term is 2 and

whose fifth term is 162.

5) The sum of a geometric series of twelve terms with common ratio 2 is 20,475.

What is the first term?

6) The sum of a geometric series of eight terms with common ratio 3 is 6,560.

What is the first term?

7) You receive an email asking you to forward it to 4 other people to ensure prosperity.

Assuming that no one person breaks the chain and there are no duplications among

the recipients, how many emails will have been received and sent after 8 email

generations, including yours?

8) A blacksmith agreed to shoe a horse on the condition that he would be paid one

cent for the first nail, two cents for the second nail, four cents for the third, and so on.

If each shoe requires eight nails, how much will the blacksmith receive for shoeing the

horse?

14

SIGMA NOTATION (DAY 6)

Warm-Up: A pile of bricks has 85 bricks in the bottom row, 81 bricks in the second row up,

77 in the third, and so on up to the top row that contains only 1 brick. How

many bricks are in the pile?

A series can be represented in a compact form, called ___________________notation or

_________________ notation.

The Greek capital letter, ,called ___________, is used to indicate a sum.

Evaluate the following:

1)

4

1j

2j 2)

7

3m

)1m2(

3) 7 +

8

5k

2 )1k( 4)

5

2k

)xk(34

5)

3

1k )2k(k

1

SIGMA (SUMMATION) NOTATION:

15

6)

500

1k

)5k3(2

1 7)

12

3j

)3j4(

8) Write the sum of the first 25 positive odd integers in sigma notation.

9) Use sigma notation to express the series 7 + 14 + 21 + 28 + … + 105.

Using sigma notation, write an expression that indicates the sum. Find the sum.

10) 2 + 4 + 6 + … + 198 = ? 11) 2 + 4 + 8 + … + 256 = ?

12) 98 + 198 + 298 + … + 1498 = ?

If the range from the first term to the last term is too large, or to do a check:

To get the summation of the sequence: PRESS

2nd STAT MATH SUM 2nd STAT OPSSEQ

You must type in and write down on your paper:

SUM(SEQ(formula, x, lower bound, upper bound))