1. 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?

2. 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 = d = a102 = 5 + (102 – 1)(8) = = 813

3. 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 = a1 + (9 – 1)d
Solve the system: a1 + 4d = 24
a1 + 8d = 40
– 4d = – = a1 + (4)4
d = a1 = 8
Explicit formula: an = 8 + (n – 1)(4)
or an = 4 + 4n
sequence
continuous
function – – –

4. 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

5. 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

6. 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

7. Homework *There are several word problems in the homework –
Pg #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!