Topic 5: Sequences and Series

Opening Problem Vicki has 30 days to train for a swimming competition. She swims 20 laps the first day, then each day after that she swims two more laps than the previous day. So she swims 22 laps the second day, 23 laps the third day and so on. How many laps does she swim on the tenth day? How many laps on the final day? How many laps does she swim total?

5A: TLW identify a sequence and apply them. Why look at a sequence? Recognize a pattern Describe a pattern Continue the pattern

Number sequence This is a list of numbers that follow a pattern Members/Terms– individual numbers in the sequence What’s the pattern? 3, 7, 11, 15, . . . 1st term = 3 2nd term = 7 3rd term = 11

Example Describe the sequence: 14, 17, 20, 23, . . . Write the next two terms: Try p. 128 #1 all and 2a, b, c

5B: TLW find the general term of a number sequence. General Term (nth term): the formula that creates the pattern. List the terms: Use words to describe the sequence.

Explicit formula 𝑢 𝑛 =𝑛+2 𝑢 1 = 𝑢 2 = 𝑢 3 =

Graphical Representation Do not join the dots! Why?

The general term Represented by symbol with subscript 𝑢 𝑛 or 𝑇 𝑛 𝑢 𝑛 : the sequence created by using 𝑢 𝑛 as the nth term. Function Notation: 𝑛→ 𝑢 𝑛 , 𝑛∈ ℤ + Try p. 130 #1, 6

Fibonacci sequence O, 1, 1, 2, 3, 5, 8, 13, 21, . . . What’s the pattern? There are examples of the sequence in nature Sunflower seeds spiral out Roots on trees divide following the pattern

5C TLW identify arithmetic sequences and apply them. The terms differ from each other by the same number Arithmetic progression 𝑢 𝑛 is arithmetic if and only if 𝑢 𝑛+1 − 𝑢 𝑛 =𝑑

Why Arithmetic? a, b and c are consecutive terms b – a = c - b

Formula 𝑢 𝑛 = 𝑢 1 + 𝑛−1 𝑑 N is always a positive integer.

Example Show that 2, 9, 16, 23, 30, . . . is arithmetic. Write the general term: What’s 𝑢 100 Is 828 a member of the sequence? Is 2341 a member of the sequence?

Example Find k given that 3k+1, k and -3 are consecutive terms of an arithmetic sequence.

Try these P. 132-133 #1-8

Some more 5C and using your gdc Find the general term 𝑢 𝑛 for an arithmetic sequence with 𝑢 3 =8 and 𝑢 8 =−17.

Example Insert 4 numbers between 3 and 13 so that all 6 numbers are in an arithmetic sequence.

Try these P. 133-134 #9-12

Example Ryan is a cartoonist. His comic strip has just been bought by a newspaper, so he sends them 28 comic strips he has drawn so far. Each week he mails 3 more. Find the total number of strips after 1, 2, 3, and 4 weeks. Show the total number after n weeks from an arithmetic sequence. How many comic strips in 15 weeks? When does he send his 120th comic strip? (What am I solving for?)

Try P. 135 #13-15

5D: Tlw identify geometric sequence and apply them. Multiply by same non-zero constant to term to get the next term. Ex: 2, 10, 50, 250, . . .

Geometric sequence 𝑢 𝑛 is a geometric ⟺ 𝑢 𝑛+1 𝑢 𝑛 =𝑟 Why geometric? 𝑏 𝑎 = 𝑐 𝑏

Formula 𝑢 𝑛 = 𝑢 1 𝑟 𝑛−1 Show sequence is geometric: 8, 4, 2, 1, ½ , . . . Find 𝑢 𝑛

try P.136 #1, 4, 5, 6, 7

example k-1, 2k and 21 – k are consecutive terms in a geometric sequence. Find k.

Try p. 138 #8 a and c

Example Find the first terms of the sequence 6, 6 2 , 12, 12 2 . . . Which exceeds 1400.

try P.138 #10

Geometric sequence problems Models growth and decay Use Solver on calculator

example The initial population of rabbits on a farm was 50. The population increased 7% each week. How many rabbits were present after: 15 weeks 3o weeks How long would it take for the population to reach 500?

Example A film club initially had 300 members. However, its a membership has since decreased by 6% each year. How many members did the club have after 5 years? How long does it take for the number of members to drop to 150? Increase: change % to a decimal and add 1. Decrease: change % to a decimal and subtract from 1.

5E TLW identify a series and apply them. Series- addition of the terms of a sequence The corresponding series of 𝑢 𝑛 is 𝑢 1 + 𝑢 2 + 𝑢 3 + . . . The sum of a series: 𝑆 𝑛 = 𝑎 1+ 𝑎 2 + 𝑎 3+…+ 𝑎 𝑛 21, 23, 25, 27, . . . ,49 is a sequence 21 +23+25+27+. . . +49 is a series

Sum of an arithmetic series 𝑆 𝑛 = 𝑛 2 𝑢 1 + 𝑢 𝑛 𝑆 𝑛 = 𝑛 2 2 𝑢 1 + 𝑛−1 𝑑 Depending on what information you are given decides which formula to use.

Example Find the sum of 4+7+10+13+… to 50 terms.

Example Sum -6+1+8+15+…+141

Try these P. 141 #1-3, 4a-d, 5-8

More 5e An arithmetic sequence has the first term 8 and a common difference of 2. The sum of the terms of the sequence is 170. Find the number of terms in the sequence.

TrY these P. 142 #10-13

Geometric series It is the addition of the successive terms of a geometric sequence. 1, 2, 4, 8, 16, …, 1024 is a geometric sequence 1+2+4+8+16+ . . . + 1024 is a geometric series Finite geometric series– adding first n terms of a geometric sequence Infinite geometric series– adding all of the terms in a geometric sequence which goes on and on forever

Sum of a finite geometric series 𝑆 𝑛 = 𝑢 1 + 𝑢 1 𝑟+ 𝑢 1 𝑟 2 + 𝑢 1 𝑟 3 +…+ 𝑢 1 𝑟 𝑛−1 𝑆 𝑛 = 𝑢 1 𝑟 𝑛 −1 𝑟−1 = 𝑆 𝑛 = 𝑢 1 1− 𝑟 𝑛 1−𝑟 , r≠1

Example Find the sum of 2 + 6 + 18 + 54 + . . . + n to 12 terms.

Example Again? Find a formula for 𝑆 𝑛 for the first n terms of 9−3+1− 1 3 + . . .

Try these P. 144 #1, 2a – d, 4

GDC and a Geometric series A geometric sequence has first term 5 and a common ratio 2. The sum of the first n terms of the sequence is 635. Find n.

Try these P. 145 #5 and 6 Change #6c to -319.9

Theory of knowledge Conjecture: statements we believe are true but not yet proven. Go to page 145

Suppose we have n points around a circle such that we connect each point with every other point, no three lines intersect at the same point. We count the number of regions we create.

5F: Tlw apply compound interest. Earn interest on deposit and interest already earned. Period– a certain amount of time

How does it work? Let’s say you deposit $1000 in a bank and leave it for 3 years with interest earned 10% per annum (p.a.), then how much do you have? Start: 1000 1 year: 1000 x 1.1 = 1100 2 years: 1000 x 1.1 x 1.1 =1210 3 years: 1000 x 1.1 x 1.1 x 1.1 = 1331 This is a geometric sequence with 𝑢 1 =1000 and r = 1.1 Money in account = 1000 𝑥 1.1 𝑛

Compound interest formula 𝐹𝑉=𝑃𝑉 𝑥 1+ 𝑟 100 𝑛

Example, it’s about time. $5000 is invested for 4 years at 7% p.a. compound interest, compounded annually. What will it amount to at the end of this period? Give your answer to the nearest cent.

Different compounding periods Interest is not always earned yearly. 𝐹𝑉=𝑃𝑉 𝑥 1+ 𝑟 100𝑘 𝑘𝑛 k = # times compounded per year Half-yearly = 2 Quarterly = 4 Monthly = 12 Weekly = 52 Daily = 365

Example Calculate the final balance of a $10 000 investment at 6% p.a. where interest is compounded quarterly over 2 years.

Interest earned Interest = FV – PV Just finds how much interest has been earned.

Example How much interest is earned if €8800 is placed in an account that pays 4 ½% p.a. compounded monthly for 3 ½ years?

Try These P. 148 #1-6

GDC and Compound interest problems 𝐴𝑃𝑃𝑆→𝐹𝑖𝑛𝑎𝑛𝑐𝑒 →𝑇𝑉𝑀 𝑆𝑜𝑙𝑣𝑒𝑟 N = # of time periods I% = interest PV = present value (in terms of money going out or in) FV = future value P/Y = # payments per year C/Y = # compoundings per year PMT = END BEGIN is when the payment is made.

Let’s Try It out. Holly invests £15 000 in an account that pays 4.25% p.a. compounded monthly for 5 years. How much is her investment worth at the end of 5 years?

You need to know… On an IB exam if you write the following for work on the previous problem then you will be awarded marks for method. N = 60 I = 4.25 PV = -15 000 C/Y = 12 FV = 18, 544.53

Example with the GDC How much does Helena need to deposit into an account to collect $50 000 at the end of 3 years if the account is paying 5.2% p.a. compounded quarterly?

Try These P. 151 #1-7

Another example, but this one is different HOW LONG must Magnus invest £4000 at 6.45% p.a. compounded half-yearly for it to amount to £10 000?

Yet another example Iman deposits $5000 in an account that compounds interest monthly. 2.5 years later the account totals $6000. What is the annual rate of interest?

Try These P. 152 #11-13

5G: Tlw apply depreciation. Depreciation is a loss in value of an item over time. What are things that you know that depreciate?

How does that work? Suppose a truck is bought at $36 000 and depreciates 25% per year. The truck is only worth 75% of the original value. Year 1: 36 000 x .75 = 27 000 Year 2: 36 000 x .75 x .75 = 20250 Year 3: 36 000 x .75 x .75 x .75 = 15188 N years: 36 000 𝑥 .75 𝑛

Depreciation The annual multiplier is 1+ 𝑟 100 , r is negative! 𝐹𝑉=𝑃𝑉 𝑥 1+ 𝑟 100 𝑛 Where have we seen this before?

example An industrial washer was purchased for £2400 and depreciated 15% each year. Find its value after 6 years. How much value has it lost?

The final example A vending machine is bought for $15 000 is sold 3 years later for $9540. Calculate its annual rate of depreciation.

Assignment P. 152 #1-5