Arithmetic sequences

Arithmetic sequences What do these sequences have in common? 3, 6, 9, 12, 15 … 5, 7, 9, 11, 13 … 2, 7, 12, 17, 22… add 3 add 2 add 5 8, 11, 14, 17, 20, … 100, 104, 108, 112, 116… add 3 add 4 All these sequences are made by adding on the same number each time. They are arithmetic sequences. Teacher notes An arithmetic sequence is also called an arithmetic progression or a linear sequence. Ask students if they can write the recursive definitions for the sequences shown on this slide. This will help them see where the general form of the recursive definition for an arithmetic sequence comes from. The recursive definitions are: 3, 6, 9, 12, 15… a1 = 3, an + 1 = an + 3 5, 7, 9, 11, 13…a1 = 5, an + 1 = an + 2 2, 7, 12, 17, 22… a1 = 2, an + 1 = an + 5 8, 11, 14, 17, 20… a1 = 8, an + 1 = an + 3 100, 104, 108, 112, 116… a1 = 100, an + 1 = an + 4 Mathematical Practices 7) Look for and make use of structure. Students should be able to see that the five sequences on the slide have a similar form, perceiving both the pattern within the sequence (for example, that 3, 6, 9, 12, 15… increases by adding 3 between each term) and the larger pattern (that all sequences increase by adding a standard value between each term). 8) Look for and express regularity in repeated reasoning. Students may notice that since each sequence has a similar structure, they should be able to construct a general definition for arithmetic sequences. The difference between any two consecutive terms in an arithmetic sequence is called the common difference, d.

Recursive definition Each term in a sequence is identified by its position in the sequence. The first term is a1, because it is in position 1. The term in position n, where n is a natural number, is called an. An arithmetic sequence can be defined recursively by the formula: Teacher notes Check that students understand why the first term needs to be given by asking for examples of sequences where the formula for the recursive definition is the same but the sequences are different. One easy pair of examples is the even and odd numbers. Both of these are arithmetic sequences defined recursively with the formula an+1 = an + 2, but the even numbers have the first term a1 = 0 and the odd numbers have the first term a1 = 1. an+1 = an + d The value of the first term needs to be given as well, so that the definition is unique.

The general term How do you find the general term of an arithmetic sequence? 8, 11, 14, 17, 20… + 3 + 3 + 3 + 3 This is an arithmetic sequence with 3 as the common difference. We could write this sequence as: 8, 8 + 3, 8 + 3 + 3, 8 + 3 + 3 + 3, … , 8 + 3 + … + 3, … Mathematical Practices 8) Look for and express regularity in repeated reasoning. Students should see that repeatedly adding on 3 to generate the next term in the sequence can be simplified to give an expression for the general term in the sequence. or n – 1 times 8, 8 + 3, 8 + (2 × 3), 8 + (3 × 3), … , 8 + ((n – 1) × 3), … The general term of this sequence is 8 + 3(n – 1)

Explicit formula If we call the first term of an arithmetic sequence a1 and the common difference d we can write a general arithmetic sequence as: 1st 2nd 3rd 4th 5th a1, a1 + d, a1 + 2d, a1 + 3d, a1 + 4d, … + d + d + d + d The explicit formula for an arithmetic sequence with common difference d is given by an = a1 + (n – 1)d

Deriving the general equation We can derive the general form of the explicit formula for an arithmetic sequence with common difference d from the equation of a line. The point–slope form of the equation of a line is: y – y1 = m(x – x1) Write n for x, an for y, and d for m: an – y1 = d(n – x1) Use the first term in the sequence, so x1 = 1 and y1 = a1: an – a1 = d(n – 1) Rearrange to isolate an: an = a1 + d(n – 1)

Car hire problems The cost of hiring a car from Carrie’s Cars depends on the number of days the car is needed for: Days Cost 1 2 3 4 5 $60 $80 $100 $120 $140 Find the explicit formula and use it to solve these problems: Teacher notes The solutions are shown on the next slide. Mathematical Practices 4) Model with mathematics. Students should model the cost of car hire using an arithmetic sequence, representing the cost as the terms in the sequence and the number of days as the position in the sequence. They should analyze this sequence mathematically, finding the explicit formula for the nth term in the sequence and using this to solve the given questions. Students should interpret their results in the context, noticing that car hire is by the day, so n can only take integer values. Photo Credit: © MANDY GODBEHEAR, Shutterstock.com 2012 1) What is the cost for 6 days? 2) How much will it cost for 2 weeks car hire? 3) How many days can Lois hire a car for if she has $250?

Car hire solutions The car hire cost is an arithmetic sequence with common difference 20. 60, 80, 100, 120, 140, … Find the explicit formula first: an = a + (n – 1)d = 60 + (n – 1)20 = 40 + 20n 1) The cost for 6 days is the 6th term: 140 + 20 = $160. Mathematical Practices 4) Model with mathematics. Students should model the cost of car hire using an arithmetic sequence, representing the cost as the terms in the sequence and the number of days as the position in the sequence. They should analyze this sequence mathematically, finding the explicit formula for the nth term in the sequence and using this to solve the given questions. Students should interpret their results in the context, noticing that car hire is by the day, so n can only take integer values. Photo Credit: © MANDY GODBEHEAR, Shutterstock.com 2012 2) For 2 weeks, n = 14 so it will cost $320. 3) Lois has $250, so solve 250 = 40 + 20n to get n = 10.5. Lois can only hire the car for full days so n must be an integer less than 10.5. She can hire the car for 10 days.