Find the nth term of a sequence
Find the index of a given term of a sequence
Given a geometric series, be able to calculate the nth partial sum
Identify a geometric series as convergent or divergent.
2. Concepts and Objectives
â« Geometric Sequences and Series
â« Find the nth term of a sequence
â« Find the index of a given term of a sequence
â« Given a geometric series, be able to calculate Sn, the nth
partial sum and vice versa
â« Identify whether a geometric series converges and its
limit
3. Geometric Sequences
â« A geometric sequence is a sequence in which each term
equals a constant multiplied by the preceding term.
â« The constant for a geometric sequence is called the
common ratio, r, because the ratio between any two
adjacent terms equals this constant.
â« Like arithmetic sequences, formulas for calculating tn for
geometric sequences can be found by linking the term
number to the term value.
5. Geometric Sequences (cont.)
â« Consider the geometric sequence:
3, 6, 12, 24, 48, âŠ
This sequence has t1 = 3 and common ratio r = 2. Thus:
1 3t =
2 3 2t =
2
3 3 2 2 3 2t = =
3
4 3 2 2 2 3 2t = =
1
3 2n
nt â
=
6. Geometric Sequences (cont.)
â« The nth term of a geometric sequence equals the first
term multiplied by (n â 1) common ratios. That is,
â« A geometric sequence is actually just an example of an
exponential function. The only difference is that the
domain of a geometric sequence is ï„ rather than all real
numbers.
1
1
n
nt t r â
=
8. Examples
1. Calculate t100 for the geometric sequence with first term
t1 = 35 and common ratio r = 1.05.
( )( )100 1
100 35 1.05t â
=
( )( )99
35 1.05 4383.375262= =
10. Examples
2. A geometric sequence has t1 = 17 and r = 2. If
tn = 34816, find n.
To solve for n, we will take the log of each side:
( )( )1
34816 17 2nâ
=
1
2048 2nâ
=
1
log2048 log2nâ
=
( )log2048 1 log2n= â
log2048
1
log2
n= â
11 1n= â
12n =
11. Geometric Series
â« If we wanted the sum of the first 100 terms of our first
geometric sequence, we could write it as follows:
â« Now, suppose we multiplied both sides of this equation
by â2, or the opposite of our common ratio, and added
the two equations together:
= + + + + + +2 3 98 99
100 3 3 2 3 2 3 2 ... 3 2 3 2S
= + + + + + +
â = â â â â â â â
2 3 98 99
100
2 3 98 99 100
100
3 3 2 3 2 3 2 ... 3 2 3 2
2 3 2 3 2 3 2 ... 3 2 3 2 3 2
S
S
13. Geometric Series (cont.)
â« The nth partial sum of a geometric series is given by the
formula
â« For some reason, Iâm always tempted to try to factor the
fraction further. It doesnât factor.
ïŠ ï¶â
= ï§ ï·
âïš ïž
1
1
1
n
n
r
S t
r
15. Geometric Series
â« Example: Find S34 for the geometric series with t1 = 7
and r = 1.03.
Using the formula, we have:
ïŠ ï¶â
= ï§ ï·
âïš ïž
34
34
1 1.03
7
1 1.03
S
ï» 404.111
16. Geometric Series
â« Example: 50238.14 is the approximate value of a partial
sum in the geometric series with t1 = 150 and r = 1.04.
Which term is it?
17. Geometric Series
â« Example: 50238.14 is the approximate value of a partial
sum in the geometric series with t1 = 150 and r = 1.04.
Which term is it?
ïŠ ï¶â
= ï§ ï·
âïš ïž
1
1
1
n
n
r
S t
r
ïŠ ï¶â
= ï§ ï·
âïš ïž
1 1.04
50238.14 150
1 1.04
n
( )( )â
= â
50238.14 0.04
1 1.04
150
n
=1.04 14.39683...n
1â1.04 = â0.04
18. Geometric Series
(cont.) Taking the log of each side gets n out of the
exponent.
So n = 68. (n always has to be a positive integer.)
=log1.04 log14.39683...n
=log1.04 log14.39683...n
= =
log14.39683...
68.000001...
log1.04
n
19. Convergent Geometric Series
â« It should be obvious that the partial sums of a geometric
sequence such as the last example will continue to
increase as n increases.
â« Now, letâs look at a different sequence:
The first six partial sums would look like:
1 1 1 1
2, 1, , , , , ...
2 4 8 16
=1 2S =2 3S =3
7
2
S =4
15
4
S =5
31
8
S =6
63
16
S
21. Convergent Geometric Series
â« With this rewritten formula, we can see that as n
increases, (œ)n-2 gets closer and closer to 0. (Check out
the value of œ raised to larger and larger powers.)
â« Therefore, we say that the limit of Sn as n increases
without bound (or approaches infinity) is 4 or
â« In order for the common ratio term to go to 0 as n
increases, the denominator of the partial sums formula
must be a proper fraction. That is, . This is called a
convergent geometric series. A series that does not
converge diverges.
âï„
=lim 4n
n
S
ïŒ1r
22. Convergent Geometric Series
â« The formula for the sum of a convergent geometric
series is
Example: In our previous sequence, t1 = 2 and r = œ:
= ïŒ
â
1
, where 1
1
t
S r
r
= = =
â
2 2
4
1 1
1
2 2
S