Aim: What is the sequence? Do Now: If f (n) = 3n - 2 find f (1), f (2), f (3), f (4). f (1) = 3(1) - 2 = 1 f (2) = 3(2) - 2 = 4 f (3) = 3(3) - 2 = 7 f (4) = 3(4) - 2 = 10 p.250 # 8,12,14,16,22,24,26,32,34,36

The numbers 1,4,7,10 form a sequence. A sequence is a set of numbers written in a given order. Each number in a sequence is called term. An infinite sequence is a function whose domain is the set of ALL positive integers. A finite sequence is a function whose domain is only the first n positive integers. We can present f(n) = 3n – 2 by a different notation as an = 3n – 2, that is the formula allows any term of a sequence, except the first, to be computed from the previous term is called recursive definition.

an represents the nth term of the sequence a1 is the first term, a2 is the second term and so on. a1,a2,a3,a4,···,an How do we write the terms from a given rule? an = 3n – 2 a1 = 3(1) – 2 = 1 a2 = 3(2) – 2 = 4 a3 = 3(3) – 2 = 7 a4 = 3(4) – 2 = 10 · · · · · · · · · · ·

Write the first six terms of the sequence an = 2n + 3. 1st term a 2 = 2(2) + 3 = 7 2nd term a 3 = 2(3) + 3 = 9 3rd term a 4 = 2(4) + 3 = 11 4th term a 5 = 2(5) + 3 = 13 5th term a 6 = 2(6) + 3 = 15 6th term

a1 = (–2) 1 – 1 = 1 a2 = (–2) 2 – 1 = –2 a3 = (–2) 3 – 1 = 4 Write the first six terms of the sequence an = (–2) n – 1 . a1 = (–2) 1 – 1 = 1 1st term a2 = (–2) 2 – 1 = –2 2nd term a3 = (–2) 3 – 1 = 4 3rd term a4 = (–2) 4 – 1 = – 8 4th term a5 = (–2) 5 – 1 = 16 5th term a6 = (–2) 6 – 1 = – 32 6th term

Writing Rules for Sequences If the terms of a sequence have a recognizable pattern, then we may be able to write a rule for the n th term of the sequence. Describe the pattern, write the next term, and write a rule for the nth term of the sequence

A rule for the nth term is an = 1 2 3 4 1 243 - 5 1 3  , 9 27 81 terms rewrite terms 1 3 - 4 , 2 1 3 - 5 1 3 - A rule for the nth term is an = n

Describe the pattern, write the next term, and write a rule for the nth term of the sequence. 2, 6, 12 , 20,…. n 1 2 3 4 5 30 terms 2 6 12 20 rewrite terms 1(1 +1) 2(2 +1) 3(3 +1) 4(4 +1) 5(5 +1) A rule for the nth term is an = n (n+1).

Write the first five terms of a) an = 5n + 2 b) 2.Write the general formula for the sequence 12, 6, 3, 1.5,… 3. Write the first five terms of a1 = 20, an = an -1 – 4