Coordinate Algebra
Geometric Sequences Day 2 Notes
Date: _______
COMPLETED
Writing an Explicit Rule for Geometric Sequences
We discussed how to write an explicit/closed form rule for an geometric sequence when given a finite or
infinite list of data points. Now we will discuss how to write the rule using a specific term and the common
ratio.
Ex1) One term of a geometric sequence is a4 = 864. The common ratio is r = -6.
Solve for a1:
an = a1rn – 1
a4 = a1r4 – 1
864 = a1(-6)3
864 = a1(-216)
Divide both sides by (-216)
a1 = -4
Substitute into original formula to get the explicit rule:
an = a1rn – 1
an = -4(-6)n – 1
Ex2) Write a rule for the nth of the geometric sequence:
an = a1rn – 1
a6 = a1r6 – 1
-64 = a1(2)5
-64 = a1(32)
a1 = -2
r = 2, a6 = -64
Divide both sides by (32)
Substitute into original formula to get the explicit rule:
an = a1rn – 1
an = -2(2)n – 1
Ex3) Write a rule for the nth of the geometric sequence:
an = a1rn – 1
a5 = a1r5 – 1
-256 = a1(4)4
-256 = a1(256)
a1 = -1
r = 4, a5 = -256
Divide both sides by (256)
Substitute into original formula to get the explicit rule:
an = a1rn – 1
an = -1(2)n – 1 OR an = -2n – 1
Try on your own:
Write a rule for the nth of the geometric sequence.
1) r = 4, a4 = 192
an = a1rn – 1
a4 = a1r4 – 1
192 = a1(4)3
192 = a1(64)
a1 = 3
an = a1rn – 1
an = 3(4)n – 1
2) r = -3, a5 = 324
an = a1rn – 1
a5 = a1r5 – 1
324 = a1(-3)4
324 = a1(81)
a1 = 4
an = a1rn – 1
an = 4(-3)n – 1
Writing a Geometric Sequence as a Recursive Formula
A recursive definition gives you the first term and a rule for how the nth term relates to the (n–1)th term.
Sometimes to write a recursive formula for a geometric sequence you must first write the explicit formula.
Ex4) Write the recursive definition given the explicit formula of a geometric sequence: an = 4(-3)n – 1
an = 4(-3)n – 1
First term (a1)
Common ratio (r)
a1 = 4, an = (-3)an – 1
Ex5) Find the recursive formula for the sequence: 3, -6, 12, -24, …
a1 = 3; r = -2
a1 = 3, an = (-2)an – 1
Ex6) Find the recursive formula for the sequence: 2, 6, 18, 24, …
a1 = 2; r = 3
a1 = 2, an = (3)an – 1