Lesson 13 – 2 Arithmetic & Geometric Sequences Pg 687   #1–3, 11, 13, 14, 15, 18, 20, 23–25, 28–30, 32, 34, 35, 38​ Lesson 13 – 2 Arithmetic & Geometric Sequences Pre-calculus Objective: - Arithmetic and Geometric Means - Arithmetic or Geometric Sequence?

Arithmetic sequence (defined recursively) A sequence a1, a2, a3, … if there is a constant d for which an = an–1 + d for n > 1 (defined explicitly) the general term is an = a1 + (n – 1)d d is the common difference d = an – an–1 Ex 1) Determine if the sequence is arithmetic. If yes, name the common difference. 20, 12, 4, –4, –12, … yes d = –8 9.3, 9.9, 10.5, 11.1, 11.7, … yes d = 0.6 Ex 2) Create your own arithmetic sequence with common difference of –1.5 (share a few together) Ex 3) Find the 102nd term of the sequence 5, 13, 21, 29, … a1 = 5 d = 8 a102 = 5 + (102 – 1)(8) = 5 + 808 = 813

The graph of a sequence is a set of points – NOT a continuous curve If we know two terms of a sequence, we can find a formula. Ex 4) In an arithmetic sequence, a5 = 24 and a9 = 40. Find the explicit formula. (write what we know) 24 = a1 + (5 – 1)d 40 = a1 + (9 – 1)d Solve the system: a1 + 4d = 24 a1 + 8d = 40 – 4d = –16 24 = a1 + (4)4 d = 4 a1 = 8 Explicit formula: an = 8 + (n – 1)(4) or an = 4 + 4n sequence continuous function – – –

If a1, a2, a3, …, ak–1, ak is an arithmetic sequence, then a2, a3, …, ak–1 are arithmetic means between a1 and ak. Ex 5) Find 3 arithmetic means between 9 and 29. 9, ___ , ___ , ___ , 29 14 19 24 *this is a stream-lined way to solve* last – first # of commas Geometric Sequence (defined recursively) A sequence a1, a2, a3, …if there is a constant r for which an = an–1 · r for n > 1 (defined explicitly) the general term is an = a1rn–1 r is the common ratio

Ex 6) Create your own geometric sequence with common ratio r = –2. (share please) Ex 7) Find an explicit formula for the geometric sequence 4, 20, 100, 500, … and use it to find the ninth term. a1 = 4

If a1, a2, a3, …, ak–1, ak is a geometric sequence, then a2, a3, …, ak–1 are called geometric means between a1 and ak. Ex 8) Locate 3 geometric means between 4 and 324. 4, ___ , ___ , ___ , 324 or 4, ___ , ___ , ___ , 324 12 36 108 *stream-lined way to solve* –12 36 –108 # of commas r = A single geometric mean is called the geometric mean. (the signs must be the same) Ex 9) Find the mean proportional m (if it exists) between: a) –42 and –378 b) 1 and –16 DNE

Homework *There are several word problems in the homework – Pg 687 #1–3, 11, 13, 14, 15, 18, 20, 23–25, 28–30, 32, 34, 35, 38 *There are several word problems in the homework – just make the sequence and apply the rules!