2. Aim: Arithmetic Sequence Course: Math Literacy Do Now: Aim: What is an arithmetic sequence? Write the first five terms of the sequence where rule for the n th term is represented by n + 2 positive integers terms of sequence n + 2 6 543 7 recursive definition?

3. Aim: Arithmetic Sequence Course: Math Literacy Definition of Arithmetic Sequence A sequence is arithmetic if the differences between consecutive terms are the same. Sequence a 1, a 2, a 3, a 4,..... a n,... is arithmetic if there is a number d such that a 2 – a 1 = d, a 3 – a 2 = d, a 4 – a 3 = d, etc. The number d is the common difference on the arithmetic sequence. Each term after the first is the sum of the preceding term and a constant, c. 7, 11, 15, 19,.... 4 4 4 4 = d 2, -3, -8, -13,.... -5 -5 -5 -5 = d 4n + 3,... 7 – 5n,... finite infinite 1 st termn th term

4. Aim: Arithmetic Sequence Course: Math Literacy The n th Term of an Arithmetic Sequence The n th term of an arithmetic sequence has the form a n = dn + c where d is the common difference between consecutive terms of the sequence and c = a 1 – d Find a formula for the n th term of the arithmetic sequence whose common difference is 3 and whose first term is 2. a 1 = 2 a n = dn + c a n = 3n – 1 2 = 3(1) + ca n = dn + cc = -1 2, 5, 8, 11, 14,..., 3n – 1,... An alternative form of the nth term is a n = a 1 + (n – 1)d 7, 11, 15, 19,.... 4 4 4 4 = d 4n + 3,...

5. Aim: Arithmetic Sequence Course: Math Literacy Recursive Formula for Arithmetic Sequence Recursive definition of a sequence – if 1 or more of the first terms are given – all other terms are defined by using the previous term(s) with a formula. positive integers terms of sequence n + 2 6 543 7 recursive definition?

6. Aim: Arithmetic Sequence Course: Math Literacy Model Problem Find the twelfth term of the arithmetic sequence 3, 8, 13, 18,.... d = 5 a n = dn + c c = a 1 – d a n = a 1 + (n – 1)d a 12 = ? a 12 = 3 + (12 – 1)5 a 12 = 3 + (11)5 = 58 Rule?a n = dn + cc = a 1 – d c = 3 – 5 = -2 a n = 5n – 2 18 = 5(4) – 2 check:

7. Aim: Arithmetic Sequence Course: Math Literacy Model Problem The fourth term of an arithmetic sequence is 20, and the 13 th term is 65. Write the first several terms of this sequence. a 13 = 9d + a 4 65 = 9d + 20 a 4 = 5(4) + c20 = 5(4) + c a n = 5n + 0 1 2 3 4 5 6 5, 10, 15, 20, 25, 30,... a n = dn + c 5 = d 0 = c a n = dn + c

8. Aim: Arithmetic Sequence Course: Math Literacy Model Problem According to the National Education Association, teachers in the United States earned an average of $30,532 in 1990. This amount has increased by approximately $1472 per year. a.Write a formula for the nth term of the arithmetic sequence that describes teachers’ average earnings n years after 1989. b.How much will US teachers earn by the year 2010? a 1 = 30,532, d = 1472 a n = a 1 + (n – 1)d a n = 30,532 + (n – 1)1472 n = 21 a 21 = 30,532 + (20)1472 = 59,972

9. Aim: Arithmetic Sequence Course: Math Literacy Model Problem According to the US Census Bureau, new one-family houses sold for an average of $159,000 in 1995. This average sales price has increased by approximately $9700 per year. a.Write a formula for the nth term of the arithmetic sequence that describes the average cost of new one-family houses n years after 1994. b.How much will new one-family houses cost, on average, by the year 2010? a 1 = 159,000, d = 9700 a n = a 1 + (n – 1)d a n = 159000 + (n – 1)9700 n = 16 a 21 = 159000 + (15)9700 = 304,500

10. Aim: Arithmetic Sequence Course: Math Literacy Arithmetic Means The terms between any two nonconsecutive terms of an arithmetic sequence are called arithmetic means. In the sequence below, 38 and 49 are the arithmetic means between 27 and 60. 5, 16, 27, 38, 49, 60 arithmetic means between 27 & 60 simple example: insert one arithmetic mean between 16 and 20 a n = a 1 + (n – 1)d 20 = 16 + (2)d d = 2 a n = a 3 = 20 a 1 = 16 (n – 1) = 3 – 1 = 2 16, 18, 20

11. Aim: Arithmetic Sequence Course: Math Literacy Model Problem Write an arithmetic sequence that has five arithmetic means between 4.9 and 2.5. a n = a 1 + (n – 1)d 2.5 = 4.9 + (6)d d = -0.4 a n = a 7 = 2.5 a 1 = 4.9 n = 7 4.9, ___, ___, ___, ___, ___, 2.5 a 2 = 4.9 + (-0.4) = 4.5 4.5 a 3 = 4.5 + (-0.4) = 4.1 a 4 = 4.1 + (-0.4) = 3.7 a 5 = 3.7 + (-0.4) = 3.3 a 6 = 3.3 + (-0.4) = 2.9 4.13.73.32.9