Geometric Sequences
Explicit Formula
𝑎𝑛 = 𝑎1 (𝑟)𝑛−1
Recursive Formula
𝑎1 = 𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚
𝑎𝑛 = 𝑟 ∗ 𝑎𝑛−1
 To find the common ratio (how you move from one number to the
next) take the 2nd number and divide it by the 1st.
 The recursive formula tells you ___________________________
 The explicit formula tells you _____________________________
 Geometric sequences MUST have a ___________ ___________
Try some:
Geometric or no?
a) 3, 9, 27, ….
1
b) 4, 1, ¼, 16…
c) 6, 7, 8,….
Find the next two terms of each sequence. Then describe the pattern.
1, 3, 9, 27, 81, _____, _____
Description: ____________
-2, 6, -18, 54, ______, ______
Description: ____________
8, 20, 50, 125, _____, _____
Description: ____________
16, 4, 1, 1/4, 1/16, _____, ______
Description: ____________
Geometric sequences are sequences of numbers where_______________________________________
____________________________________________________________________________________.
Is the sequence geometric?
9, 27, 81, ….
2, -10, 50, -250…
Common ratio: _________
Geometric?
Explicit Formula:_________
Recursive Formula: ________________
The sequence 4, 8, 16, 32…is geometric. State the
common ratio, explicit formula and the 6th term.
State the common ratio and recursive formula for
the sequence -1, 4, -16, 64…
Ratio:________
Ratio: __________
Explicit: _______________
6th term: _____________
Common ratio:_____
Recursive: _______________________
Who’s correct?
Sasha and Arnold are trying to find the recursive formula, ratio and the explicit formula for the
sequence -3, -12, -48, -192… Their answers are provided. Is either one correct? Write your response
below the work.
Sasha’s work
Common ratio: -12/-3 = 4
Explicit formula:
𝑎𝑛 = −3(4)𝑛−1
Recursive formula:
𝑎1 = −3
𝑎𝑛 = −3 ∗ 𝑎𝑛−1
Explanation:
Arnold’s work
Common ratio: -12/-3 = 4
Explicit formula:
𝑎𝑛 = 4(−3)𝑛−1
Recursive formula:
𝑎1 = −3
𝑎𝑛 = 4 ∗ 𝑎𝑛−1
Come up with your own geometric sequence. Create the recursive and explicit formulas for it.
Compare the following
𝑓(𝑥 ) = 1(−3)𝑥
1, -3, 9, -27….
1st term: __________
1st term: _____________
Ratio/base: _________
Ratio/base: ___________
Explicit formula: __________
Sequence: _____________
8, 4, 2,…
What is the ratio?
What is the base?
What is 𝑎0 ?
What is the y-intercept?
What is the explicit equation?
What is equation?
Let’s look more closely at the pattern 1, 3, 9, 27, 81… Suppose the domain is the position of a term (1, 2, 3, 4,
etc.) and the range is the term.
Make a graph of the points that are made (position, term) with the pattern.
What quadrant(s) are these points in? Why?
What kind of graph do you have?
GSE Honors Algebra 1
Unit 4 Geometric Sequences
Determine if the sequence is geometric. If it is, state the common ratio.
1)
2, 4, 8, 16, …
2) 12, 36, 108, 324, …
4) an = n2
3)
Write the equation for the nth term (an) of each geometric sequence. Make sure to use
5) 2, 10, 50, 250, ….
6) -0.25, 2, -16, 128, …
7) a3 =16, r = 4
𝒂𝒏 = 𝒂𝟏 (𝒓)𝒏−𝟏
8) 16, 4, 1, 1/4, 1/16…
Finding a term.
9) Find the 5th term when the first term is 3 and the common ratio is 2.
10) Find a9 of the sequence 5, 15, 45,…
Find an for each sequence.
11) first term is 20 and common ratio is ½.
12) 8, 2, ½, …
13) a1 = 2, n = 10 r = 3
**Find the geometric means (missing terms) of each sequence.
14) 3, _____, _____, _____, 768
15) 2, _____, _____, _____, 1250