Transcript Document

Warm Up: Multiple Choice Practice
Geometric
Sequences
Unit 6 Notes (Part 2)
AA1.CC
Review: Writing Geometric Formulas
an  a1r
n 1 𝑎𝑛 = 𝑎𝑛−1 × 𝑟
𝑎1 = 𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚
YOU NEED TWO THINGS EVERY TIME:
r (common ratio)
a1
(first term)
Ex.1 Given the first term and the common ratio of a
geometric sequence, find the explicit and recursive formulas.
a1  3
r2
a1  4
r  8
Explicit: 𝒂𝒏 = −𝟑 𝟐 𝒏−𝟏
Recursive: 𝒂𝒏 = 𝒂𝒏−𝟏 × 𝟐
𝒂𝟏 = −𝟑
𝒏−𝟏
Explicit: 𝒂𝒏 = 𝟒 −𝟖
Recursive: 𝒂𝒏 = 𝒂𝒏−𝟏 × −𝟖
𝒂𝟏 = 𝟒
Ex.2 Find the common ratio and the explicit formula:
Steps:
1. Find r
2. Identify first term
3. Plug both into the
formula
4. Can’t simplify!
1) -3, 15, -75, 375, …
𝒓 = −𝟓
𝒂𝒏 = −𝟑 −𝟓
𝒏−𝟏
2) 1, 3, 9, 27, …
𝒓 =𝟑
𝒂𝒏 = 𝟑𝒏−𝟏
3) 4, 20, 100, 500, …
𝒓 = 𝟓
𝒂𝒏 = 𝟒 𝟓
𝒏−𝟏
Ex.3 Word problem 
A chain e-mail instructs the recipient to forward the email to four more people. The table shows the
number of rounds of sending the e-mails and the
number of new e-mails generated. Write the explicit
formula of the sequence and find the next 2 terms.
# of rounds sending emails, n 1
2
3
4
# of new emails generated, an 1
4
16 64
Explicit: 𝒂𝒏 = 𝟒𝒏−𝟏
Next 2 terms: 256 and 1024
Ex.4 Given a term in a geometric sequence and the
common ratio, find the explicit formula.
Solve for the first
term and then write
the explicit formula.
−𝟏𝟎𝟎 =
−𝟏𝟎𝟎 =
−𝟏𝟎𝟎 =
−𝟒 =
𝒂𝟏 𝟓 𝟑−𝟏
𝒂𝟏 𝟓 𝟐
𝒂𝟏 × 𝟐𝟓
𝒂𝟏
an  a1r
n 1
a3  100 r  5
𝒂𝒏 = −𝟒 𝟓
𝒏−𝟏
Practice:
find the explicit formula.
a2  15
r 5
𝒂𝒏 = −𝟑 𝟓
𝒏−𝟏
a5  81
r 3
𝒂𝒏 = −𝟑
𝒏−𝟏
Ex.5 Given two terms in a geometric sequence, find
the common ratio , explicit formula, and the recursive formula.
• Find r
• Pick one term
and find the first
term
• Write the
explicit formula
and Recursive
𝒂𝟐 = 𝟔𝟎
𝒂𝟓 = 𝟕𝟓𝟎𝟎
common ratio explicit formula recursive formula
𝑛−1
𝑎𝑛 = 𝑎𝑛−1 × 5
𝑎
=
12
5
𝑛
5
𝑎1 = 12
Practice #1
find the common ratio , explicit formula, and the
recursive formula.
𝒂𝟓 = 𝟏𝟔
𝒂𝟑 = 𝟒
Practice #2
find the common ratio , explicit formula, and the
recursive formula.
𝒂𝟒 = −𝟖𝟔𝟒
𝒂𝟔 = −𝟑𝟏𝟏𝟎𝟒
Practice #3
find the common ratio , explicit formula, and the
recursive formula.
𝒂𝟏 = −𝟑
𝒂𝟑 = −𝟏𝟎𝟖
Homework:
(Part 2)
Geometric sequence worksheet
#15 – 26