UNIT 7 – DISCRETE FUNCTIONS: SEQUENCES and SERIES

Date Lesson Text TOPIC Homework

May 29 7.1

(71) 7.1

Arithmetic Sequences

TRIG IDENTITIES QUIZ

Pg. 424 # 4, 5, 6, 8 – 13, 15

WS 7.1/2 # 1, 5, 7, 8, 11, 12, 14

May 30 7.2

(72) 7.2

Geometric Sequences Pg. 430 # 5 – 8, 10 - 14

WS 7.1/2 # 2, 3, 4, 6, 9, 10, 13

May 31 7.3

(76) 7.5

Arithmetic Series

Pg. 452 # 1 – 7, 10 – 12, 15

June 3 7.4

(77) 7.6

Geometric Series

QUIZ (7.1-7.2)

Pg. 459 # 1 – 7, 9, 11, 12

June 4 7.5

(79)

Series and Sequences Work Period WS 7.6

June 5 7.6

(80)

Compound Interest and Annuities

Future Value

QUIZ (7.3-7.4)

Pg. 490 # 3, 4abd, 10, 11

Pg. 498 # 1ab, 4

Pg. 511 # 2acd, 5abd, 6

June 6 7.7

(81)

Annuities

Present Value

Pg. 521 # 4, 5c, 6, 7, 8, 10, 11

June

7/10

7.8

(82)

Review for Unit 7 Test

Pg. 468 # 3 – 5, 7 – 9, 11, 13,

14abc, 15acf, 16ac, 17, 18acf,

19ac, 20, 22ace, 23bcd Pascal

Pg. 534 # 4, 8ab, 10, 11ab, 12,

13, 14cd, 15, 17, 18

June

11

7.9

(83)

UNIT 7 TEST

June

1218 Exam Review

Pg. 206 # 1-3, 5-11, 13-20,

22-24, 26, 27, 30-32

Pg. 408 # 1-18, 20-28

Pg. 538 # 1-14

June 24

Exam – Room 117

@ 9:30

MCR3U Lesson 7.1 Arithmetic Sequences

In this unit, we must have deal. You must find all answers using algebraic methods and I will not ask you to

find ridiculous values like the 1 209 867th term of a sequence. If algebraic means are not used, the most

you can earn on any question is one mark.

Ex. 1 Determine 20t , the twentieth term of each of the following sequences:

a) 2, 5, 8, 11, … b) 21, 15, 9, 3, … c) -17, -15, -13, …

Ex. 2 Determine nt , the general term of each of the following sequences:

a) 2, 5, 8, 11, … b) 21, 15, 9, 3, … c) -17, -15, -13, …

Ex. 3 Determine nt , the general term of each of the following arithmetic sequences:

a) 16, 25, 34, 43, 52, … b) 11, 7, 3, -1, … c) ,...6

5,

2

1,

6

1

The general term of an arithmetic sequence is:

Ex. 4 How many terms are in the sequence 23, 27, 31, …, 67? (Show your work)

Ex. 5 In an arithmetic sequence 23t 7 and 13t11 . What is 28t ?

Ex. 6 Stompin’ Tom invests $300 in a GIC (Guaranteed Investment Certificate) that pays 6% simple interest

per year. When will his investment be worth $498?

Pg.424 #4, 5, 6, 8 - 13, 15

*ignore "recursive" questions!!!* WS 7.1/2 # 1, 5, 7, 8, 11, 12, 14

MCR3U Lesson 7.2 Geometric Sequences

Ex. 1 Determine 8t in each of the following sequences:

a) 2, 6, 18, 54,… b) 405, 135, 45, 15…

Ex. 2 Determine nt , the general term of each of the following sequences:

a) 2, 6, 18, 54,… b) 405, 135, 45, 5…

Ex. 3 How many terms are in each of the following geometric sequences? (show your work)

a) 3, 6, 12, … , 768 b) 9

4,...2916,8748,26244

Ex. 4 In an geometric sequence 18t 6 and 288t10 . What is 8t ? 3t ?

The general term of a geometric sequence is:

Pg. 430 # 5 - 8, 10 - 14

*ignore "recursive" questions!!!*

WS 7.1/2 # 2, 3, 4, 6, 9, 10, 13

MCR 3U Lesson 7.3 Arithmetic Series

A series is the sum of a sequence. For the sequence 1, 4, 7, 10 we say that 104 t and 22107414 S .

Ex. 1 Evaluate: 1 + 5 + 9 + 13 + 17 + 21

Ex. 2 Find 13S for 23 + 20 + 17 + 14 + …

Ex. 3 Evaluate: – 63 – 59 – 55 . . . + 17

Ex. 4 In a series, 1911 t and 6117 t . Determine 12S .

Pg. 452 #1 - 7, 10 - 12, 15

when you know the last term

MCR 3U Lesson 7.4 Geometric Series

Ex. 1 Evaluate: 4861625418626 S

Ex. 2 Evaluate: 49152 + 12288 + 3072 + … + 12

When you know the last term Ex. 3 Evaluate: ...201059 S

Ex. 4 Evaluate: ...18

1

6

1

2

17 S

Pg. 459 #1 - 7, 9, 11, 12

or 1,1

)1(

r

r

raS

n

n

1,1

r

r

atrS n

n

MCR 3U Lesson 7.5 Series and Sequences Work Period

Ex. 1 The terms given by 2x , 13 x , and 14 x form an arithmetic sequence. Find the value of x.

Ex. 2 A jogger running along a course with a slight uphill grade covers 350 m in the first minute, but due

to fatigue covers 25 m less in each succeeding minute. What distance is covered in the eighth

minute?

Ex. 3 In a geometric sequence 183 t and 14587 t . Determine r and a.

Ex. 4 In an arithmetic series 1512 t and 10515 S . Find the third term of the sequence.

Complete the rest of WS 7.5

MCR 3U WS 7.5 Series and Sequences Worksheet

1. The terms given by 2x , 13 x , and 14 x form an arithmetic sequence. Find the value of x.

2. A jogger running along a course with a slight uphill grade covers 350 m in the first minute, but due to

fatigue covers 25 m less in each succeeding minute. What distance is covered in the eighth minute?

3. In a geometric sequence 183 t and 14587 t . Determine r and a.

4. The fifth term of a geometric sequence is 27

4 and the eighth term is

729

4. What is the third term?

5. Determine y so that 2y , 105 y , and 50y form

a) an arithmetic sequence. b) a geometric sequence.

6. In an arithmetic sequence, 236 t and 4714 t . Determine d and a.

7. In an arithmetic sequence, 385 t and 10815 t . Determine 19t

8. If the second term and twelfth term of an arithmetic series are -1 and 19 respectively, find the sum of the

first thirteen terms of the series.

9. In an arithmetic series 1512 t and 10515 S . Find the third term of the sequence.

10. In an arithmetic series 46 t and 26118 S . Find the sum of the first 22 terms of the series.

11. A pile of logs has 25 logs in the bottom row, 24 logs in the next row, and so on. If there are 12 logs in the

top row, how many logs are there in the pile?

12. In a lottery there are 10 prizes. The last prize is $25 and each prize is double the preceding prize.

a) What is the top prize?

b) What is the total amount of prize money awarded?

13. Sarah is training for a marathon. This week she ran 45 km. She intends to increase her distance by 90 m each

week. How far (to the nearest kilometer) will she have run after 10 weeks?

14. Derek climbed 60 m up a cliff in the first hour of climbing. In each of the next 4 hours he climbed only

75% of the distance of the previous hour. To the nearest metre, how far had he climbed after 5 hours?

Answers

1) 3 2) 175m 3) 2,3 4) 3

4 5) a) -9 b) 0 or

3

19 6) 3, 8 7) -136

8) 117 9) -3 10) 451 11) 259 12a) $12,800 12b) $25,575 13) 454 km 14) 183 m

MCR 3U Lesson 7.6 Compound Interest and Annuities

COMPOUND INTEREST

Ex. 1 Calculate the accumulated amount of:

a) $100 invested for 6 years at 3.5%/a compounded annually.

b) $2000 invested for 7 years at 6%/a c) $4500 invested for 5 years at 7%/a

compounded annually. compounded annually.

d) $100 invested for 6 years at 3.5%/a e) $2000 invested for 7 years at 6%/a.

compounded semi-annually. compounded monthly.

f) $4500 invested for 5 years at 7%/a

compounded quarterly.

Ex. 2 How much must you invest today in order to have $10000 in 5 years if money is worth

4.5%/a compounded semi-annually?

ANNUITIES (FUTURE VALUE)

Ex. 1 Calculate the accumulated amount of:

a) $100 invested at the end of each year for 30 years at 8% compounded annually.

b) $20 invested at the end of each month for 25 years at 6% compounded monthly.

Ex. 2 Jane wants to have $3000 in one year to take a trip to Spain. She has found an investment account at

her bank that will pay 7.2% per year compounded monthly. What monthly payments must she make to

reach her goal?

Ex. In the annuity in Ex. 1, $450 is deposited at the end of each quarter for 1.5 years at 10% per year

compounded quarterly. Use the TVM solver to determine the amount of the annuity?

Pg. 490 # 3, 4abd, 10, 11

Pg. 498 # 1ab, 4

Pg. 511 # 2acd, 5abd, 6

If using the TI-83, press [2nd

] [x-1

] [1]

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PMT: END BEGIN

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Using the TI-83 to Solve Annuity Problems

Ex. 1 Amira and Bethany are twins. They save for retirement as shown below.

Starting at age 25, Amira deposits $1000 at the end of each year for 40 years.

Starting at age 40, Bethany deposits $2000 at the end of each year for 25 years.

Suppose that each annuity earns 8% per year compounded annually. Who will have the greater

amount at retirement?

Amira Bethany

Ex. 2 Habeeba wants to save $3000 for the Japan trip in 3 years. What regular deposit should she make

at the end of every 6 months in an account that earns 6% per year compounded semi-annually?

Ex. 3 Gurnoor is 8 years old. His parents would like to have $40 000 to pay for his education when he

leaves for university in 10 years. They plan to make monthly payments of $250 for the next 10

years. What interest rate must they receive to achieve their goal? State your answer to

1 decimal place.

N =

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N =

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N =

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MCR 3U Lesson 7.7 Present Value of an Annuity

compounding periods.

Ex. 1 Zeena buys a new car. She has to make a $2000 downpayment and then to make

monthly payments of $450 for 5 years. What is the cost of the car if the

interest rate is 3.6%/a compounded monthly.

Ex. 2 You borrow $3000 to buy a new 3D TV. The financing department suggests you

can pay for it with monthly payments over two years. Their interest rate is 39%

compounded monthly. Determine the monthly payment. Should you do this?

Justify your reasoning.

Pg. 521 # 4, 5c, 6, 7, 8, 10, 11

Using the TI-83 to Solve Annuity Problems

Ex. 4 Andrea plans to retire at age 55. She would like to have enough money saved in her account so she

can withdraw $7500 every 3 months for 30 years, starting 3 months after she retires. How much

must she deposit at retirement at 9% per year compounded quarterly to provide for the annuity?

Ex. 5 Allison plans to buy a car. She can afford monthly payments of $300. The car dealer offers her a

loan at 6.9% per year compounded monthly, for 4 years. The first payment will be made 1 month

from when she buys the car. How much can she afford to borrow?

Ex. 6 How many monthly payments of $125 are needed to save $5000 if invested at 4.5% compounded

monthly?

Annually 1 time per year i = annual interest rate n = number of years

Semi-Annually 2 times per year i = annual interest rate 2 n = number of years x 2

Quarterly 4 times per year i = annual interest rate 4 n = number of years x 4

Monthly 12 times per year i = annual interest rate 12 n = number of years x 12

Bi-Weekly 26 times per year i = annual interest rate 26 n = number of years x 26

N =

I% =

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N =

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MCR 3U Lesson 7.8 Pascal’s Triangle and Binomial Expansion

A child’s toy called “Rockin’ Rollers” involves

dropping a marble into its top. When the

marble hits a pin it has the same chance of

going either left or right. A version of the

toy with five levels is shown at the right.

Calculate the number of paths to each bin at

the bottom.

In order to make the examples easier to explain, we're going to work with a binomial that

will contain only variables. The expression we'll use will be (x + y)

Here are the first few expansions of (x + y)n ...

(x + y)0 = 1

(x + y)1 = 1x + 1y

(x + y)2 = 1x2 + 2xy + 1y2

(x + y)3 = 1x3 + 3x2y + 3xy2 + 1y3

(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4

Look at the last line, for power n = 4. Examine the pattern carefully. First notice how the

variables appear in the answer. Each answer begins with the highest possible power of x

(whatever n was), and no y. Each term thereafter has one less power of x, and one more power

of y. It ends with the last term, which has no x and yn.

Here's the next power ...

(x + y)5 = 1x5y0 + 5x4y1 + 10x3y2 + 10x2y3 + 5x1y4 + 1x0y5

The powers of x in the answer descend from 5 down to 0.

The powers of y ascend from 0 up to 5.

(Ordinarily you wouldn't bother showing x0 or y0. We show them here just to make the pattern

clearer).

Now let's look at the pattern of the coeffients.

Here are the first few expansions again:

(x + y)0 = 1

(x + y)1 = 1x + 1y

(x + y)2 = 1x2 + 2xy + 1y2

(x + y)3 = 1x3 + 3x2y + 3xy2 +1y3

(x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4

The coeffiecients (in red) form a distinct pattern. The pattern will become more obvious when

we write them again as a symmetrical triangle:

Each row in the triangle begins and ends with a 1

To get the other numbers, you add the two numbers in the row directly above.

Do you see the pattern? The next row will begin and end with a 1.

After the first 1, the next number will be the sum of 1 and 4, which is 5,

and so on .... here's the triangle with the next row added:

You can extend Pascal's Triangle easily to as many rows as needed. To do a binomial expansion,

you need to know that the rows correspond to the power, beginning with 0.

So (x + y)0 = 1 and (x + y)1 = 1x + 1y and (x + y)2 = 1x2y0 + 2x1y1 + 1x0y2

and so on.

If we wanted the answer for (x + y)4, we'd use the coeffiecients 1, 4, 6, 4, 1 and the

pattern for powers of x and y, and we'd get:

(x + y)4 = 1x4y0 + 4x3y1 + 6x2y2 + 4x1y3 + 1x0y4

O,K., back to binomial expansions. We now want to be able to do a problem like this:

(2a3 - 5b2)4

where we aren't just dealing with x and y, but other variables with their own exponents.

The method we outlined above makes this problem a lot easier than it would have been

otherwise.

It's a power 4 question ... (2a3 - 5b2)4

If the variables were x and y, the answer would look like this: (we'll leave out the powers of 0)

(x + y)4 = 1x4 + 4x3y1 + 6x2y2 + 4x1y3 + 1y4

Now replace x and y by (2a3) and (-5b2)

= 1(2a3)4 + 4(2a3)3(-5b2)1 + 6(2a3)2(-5b2)2 + 4(2a3)1(-5b2)3 + 1(-5b2)4

and the rest is just easy algebra! Here it is:

= 1(16a12) + 4(8a9)(-5b2) + 6(4a6)(25b4) + 4(2a3)(-125b6) + 1(625b8)

= 16a12 - 160a9b2 + 600a6b4 - 1000a3b6 + 625b8

As with all new algebraic methods you are shown, you won't really learn it by watching it be

explained. You need to practice it many times to get good at it. You learn best by 'doing'.

Ex. Expand the following.

a) 33yx

b) 432 ba

Ex. Find the first four terms in the expansion of 83 2x

Pg. 466 # 1, 4abcdf, 5abcf, 10

Pascal’s Triangle 0

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Pascal’s Triangle 0

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