Geometric Sequence
A geometric sequence is a sequence in which there is a
constant common ratio (r) between one term and the next.
The first term (a) is required.
Sequence: a, ar, ar2, ar3, ...
is infinite
a, ar, ar2, ar3, ..., arn‐1 is finite
The general term can be expressed as:
tn = arn‐1; nεN
how many times we
multiplied by r
Ex. Determine the general term for the geometric sequence:
2, 6, 18, 54, 162, ...
a=2
r=3
How do you know a sequence is geometric?
Ex. Is the sequence geometric?
If so, state the common ratio.
a) 2, ‐8, 32, ‐128
c) x7, x14, x28, x56, ...
b) x, 2x, 3x, 4x, ...
d) x7, x10, x13, x16, ...
Ex. Graph the geometric sequence: 2, 6, 18, 54, 162, ...
The graph of a geometric sequence
is non‐linear and the domain is
restricted to the Natural numbers.
If r>1 , it is _____________.
If 0<r<1, it is _____________.
Ex. A geometric sequence is defined by a = 32 and r= 0.5.
Determine the position of the term with the value of 1/8.
Geometric Series = t1 + t2 + t3 + ... + tn
tn = arn‐1; nεN
The sum of an geometric sequence up to term tn is
Sn = tn+1‐t1 ; r≠1
r‐1
OR
Sn = a(rn ‐1); r≠1
r‐1
What is the sum of an infinite geometric series?
Deriving sum of geometric sequence
OOoooo...
Ahhh.... Thank you for deriving for us, Di.
Ex. Determine the sum of the finite series
400+200+100+ ... +6.25
arithmetic or geometric?
a=
r=
Sn = tn+1‐t1 ; r≠1
r‐1
OR
Sn = a(rn ‐1); r≠1
r‐1
OR
Sn = a(rn ‐1); r≠1
r‐1
a=
r=
Need to determine n of the last term.
Calculate the sum.
Sn = tn+1‐t1 ; r≠1
r‐1
Summary of Arithmetic and Geometric Sequences and Series
Type
Arithmetic
Geometric
General Term
Sum of Series
from t1 to tn
tn = a + (n‐1)d
a ‐ 1st term, t1
d ‐ common
difference
Sn = n(t1+tn)
2
OR
Sn = n [2a+(n‐1)d]
2
Sn = tn+1‐t1 ; r≠1
tn = a rn‐1
r‐1
a ‐ 1st term, t1
OR
r ‐ common
Sn = a(rn ‐1); r≠1
ratio
HW
pg. 47 #3adf, 5ac,
7ad, 11ab, 13
pg. 116 #4ac, 5ac,
6ac, 8ac,10, 23
pg. 57 #2,3,8acd,
15
pg. 123 #1, 5acd,
8, 10
r‐1
Quiz Summary
Quiz Summary ‐ Unit 6 so far
Discrete versus continuous
Sequence ­finding the pattern
­recursive form, notation
­Fibonacci
­general term, notation
­Arithmetic
­Geometric
Series
­find the sum
­Arithmetic
­Geometric