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 Find the nth term of an arithmetic sequence Find the number of terms in an arithmetic sequence Solve real-world problems involving arithmetic sequences and series. From IB SL Study Guide

Challenge – Nob’s Tricky Sequence Nob 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 can’t 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.

Make some sequences by picking four numbers that form a pattern Make 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.

How do you see this pattern growing? 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: n Draw shapes 4 and 5. How many matchsticks are in shape 10? Can you describe the pattern using algebra?

Finding the General Term Use 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.

Each day a runner trains for a 10km race Each 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 = 1000 an = 1000 + (n-1)250 = 10,000 n = 37

In 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 + 3d 16 = -2 +3d d = 6 11,998 = -2 + (n-1)6 n = 2001

Question – Finding Un Given Two Terms In 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 continued… To find Un , substitute 67 for a and 9 for d in the formula for the nth term. Un = 67 + (n – 1)9 Un = 67 + 9n – 9 Un = 9n + 58 Thus, the first three terms are 67, 76, and 85, and Un = 9n + 58.

You Should Know… A sequence is arithmetic if the difference between consecutive terms is the same An arithmetic sequence has the form: u1, u1 + d, u1 + 2d, u1 + 3d, …, u1 + (n – 1)d The common difference can be found by subtracting a term from the subsequent term: d = un + 1 – un When to use the term formula Summary slide

You should know: Textbook: Arithmetic Sequences p.155 – 159 Homework: Arithmetic Sequences

S. Aldous, A. Beetz & S. Thauvette IB DP SL Mathematics Arithmetic series S. Aldous, A. Beetz & S. Thauvette IB DP SL Mathematics

Arithmetic Series Calculate the sum of the first n terms of an arithmetic series

Challenge The 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: 312

Treasure Hunt In 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 card’s 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.

Sum of a Series Given First Terms Find the sum of the first 60 terms of the series: (a) 5 + 8 + 11 + …

Sum of a Series Given First and Last Terms Consider 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 = –10 Therefore, d = –10 and the series is arithmetic

Continued… Consider 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 = 33 Now use the appropriate formula to find the sum of the first 33 terms. S33 = –4719

Question The 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 Prepared Look for words or expressions that suggest the use of the term formula—”after the 10th month”, “in the 8th row”—and those that suggest the sum formula— ”total 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 sequence

You should know: When to use the sum formula Textbook: Arithmetic Series p.167 – 169 Homework: Arithmetic Series