Topic 1 – algebra arithmetic sequences & series

Topic 1 – algebraarithmetic sequences & series S. Aldous, A. Beetz & S. ThauvetteIB 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.

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?

Make some sequences by picking four numbers that form a pattern. Record as many as you can.

How 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?

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.

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?

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

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

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

Arithmetic series S. Aldous, A. Beetz & S. ThauvetteIB 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?

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.

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

Topic 1 – algebra arithmetic sequences & series