Click here to load reader

Topic 1 – algebra arithmetic sequences & series

  • View
    59

  • Download
    0

Embed Size (px)

DESCRIPTION

Topic 1 – algebra arithmetic sequences & series. S. Aldous, A. Beetz & S. Thauvette IB DP SL Mathematics. You Should Be Able To…. State whether a sequence is arithmetic, giving an appropriate reason Find the common difference in an arithmetic sequence - PowerPoint PPT Presentation

Text of Topic 1 – algebra arithmetic sequences & series

Academic presentation for college course (paper and pencil design)

Topic 1 algebraarithmetic sequences & seriesS. Aldous, A. Beetz & S. ThauvetteIB DP SL Mathematics1You Should Be Able ToState whether a sequence is arithmetic, giving an appropriate reasonFind the common difference in an arithmetic sequenceFind the nth term of an arithmetic sequenceFind the number of terms in an arithmetic sequenceSolve real-world problems involving arithmetic sequences and series.From IB SL Study Guide2Challenge Nobs Tricky SequenceNob Yoshigahara discovered this beautiful number sequence. Can you work out the logic behind the sequence and fill in the missing number?

Answer:

Most people who approach this problem see each number as the difference between the two numbers feeding into it. But that cant account for the 7, since 21 13 = 8.

Instead, examine the individual digits of the numbers feeding into each circle. You will find that 9, 9, 7 and 2 add up to 27, and that 4, 5, 2 and 7 total 18. Thus, the missing number can be found by adding 3, 6, 2 and 1. The missing number is 12.3Make some sequences by picking four numbers that form a pattern. Record as many as you can.

Hopefully students will make many sequences, some arithmetic, some geometric, some neither.Whole class discussion:Ask students to tell you their favourite sequence and record these on the board. Ask students to look for the rules in these sequences and describe them in words. (Example: add two each time, starting at 7.)Ask what they notice about different types of sequences.Introduce the words arithmetic and geometric to describe the sequences. We will concentrate on arithmetic sequences in this lesson.4How do you see this pattern growing?

Draw shapes 4 and 5.How many matchsticks are in shape 10?Can you describe the pattern using algebra?Start with the title question and ask students to discuss in pairs.Take responses from the class and record on the board the different ways students see the shapes growing.Point to draw out: It is useful and natural that we see things different ways. This helps us represent math in different ways.

Instead of drawing shapes 4 and 5, students could build them using popsicle sticks or toothpicks.

The subsequent questions allow the class to generalise what they see.Try to relate the way we see the growth to the algebraic representation.This is the teaching time for the general term formula. Use the same notation as in the formula booklet. First term: u1, difference: d, term number: n5Finding the General TermUse 2 pieces of paper.On one, make up a value for u1.On the other, make up a value for u4.

Swap the cards with someone else.Find the general term for the arithmetic sequence.Make sure you both agree.Students can swap by:--Mixing up all the cards in the centre of the table and then redistributing them.--Getting out of their seat to meet someone new and swqpping with them.--Etc.

This activity replaces textbook practise. Do as many as you like.6Each day a runner trains for a 10km race. On the first day she runs 1,000m, and then increases the distance by 250m each subsequent day. On which day does she run a distance of 10km in training?10km = 10,000m and will be run on the 37th day.

A1 = 1000an = 1000 + (n-1)250 = 10,000n = 377In an arithmetic sequence, the first term is 2, the fourth term is 16, and the nth term is 11,998. Find the common difference d. Find the value of n u4 = u1 + 3d16 = -2 +3dd = 6

11,998 = -2 + (n-1)6n = 20018Question Finding Un Given Two TermsIn an arithmetic sequence, U7 = 121 and U15 = 193. Find the first three terms of the sequence and Un.

Substitute know values in the formula for the nth term to write a system of equations. Then, solve the system.

Since a = 67 and d = 9, the first three terms of the sequence are 67, 76, and 85.

Finding Un Given Two Terms continuedTo find Un , substitute 67 for a and 9 for d in the formula for the nth term.

Un = 67 + (n 1)9Un = 67 + 9n 9 Un = 9n + 58

Thus, the first three terms are 67, 76, and 85, and Un = 9n + 58.

You Should KnowA sequence is arithmetic if the difference between consecutive terms is the sameAn arithmetic sequence has the form: u1, u1 + d, u1 + 2d, u1 + 3d, , u1 + (n 1)dThe common difference can be found by subtracting a term from the subsequent term:d = un + 1 un When to use the term formulaSummary slide11You should know:Textbook: Arithmetic Sequences p.155 159Homework: Arithmetic Sequences

Arithmetic seriesS. Aldous, A. Beetz & S. ThauvetteIB DP SL Mathematics13Arithmetic SeriesCalculate the sum of the first n terms of an arithmetic seriesChallengeThe top three layers of boxes in a store display are arranged as shown. If the pattern continues, and there are 12 layers in the display, what is the total number of boxes in the display?

Answer: 31215Treasure HuntIn the pod there are ten pink cards.Find any card. Note down its number.Solve the question on the card. Find the answer on another card somewhere in the pod. Note down the cards number.Continue answering questions and noting the card numbers. You should finish at the same card you started.Show your teacher the list of card numbers you visited.Students could use mini whiteboards or their notebooks for this activity.The correct order of the cards is written on the paper in the packet! Oops, need to copy it in here.16Sum of a Series Given First TermsFind the sum of the first 60 terms of the series:

(a) 5 + 8 + 11 +

Sum of a Series Given First and Last TermsConsider the series 17, 7, 3, , 303.

(a) Show that the series is arithmetic.

Show that the difference between two consecutive terms is constant. For example:7 17 = 3 7 = 10Therefore, d = 10 and the series is arithmetic ContinuedConsider the series 17, 7, 3, , 303.

(b) Find the sum of the series.

The formula for the sum of an arithmetic series requires the value of n. Use the term formula first to find n.n = 33Now use the appropriate formula to find the sum of the first 33 terms.S33 = 4719QuestionThe sum of the first five terms of an arithmetic series is 65/2. Also, five times the 7th term is the same as six times the second term. Find the first term and common difference. Question continued

Be PreparedLook for words or expressions that suggest the use of the term formulaafter the 10th month, in the 8th rowand those that suggest the sum formulatotal cost, total distance, altogether.Look for questions in which information is given about two terms. This normally suggests the formation of a pair of simultaneous equations that you will have to solve to find the first term and the common difference.The last term of a sequence can be used to find the number of terms in the sequenceYou should know:When to use the sum formula

Textbook: Arithmetic Series p.167 169Homework: Arithmetic Series