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Geometric sequence
Geometric sequence
It is a sequence of numbers where each term after the
first is found by multiplying the previous one by a fixed, non-zero number called the
common ratio.
Example:
2, 4, 8, 16, 32, 64, 128…
This sequence has a factor of 2 between each number.
Each term (except the first term) is found by multiplying the previous term by 2.
* To find the common ratio of every geometric
sequence, divide a pair of terms. It doesn't matter
which pair as long as they're right next to each
other.
*To calculate for the any term within a sequence:
𝑎𝑛 = 𝑎1 • 𝑟𝑛−1
n=position of the term in the sequence a1= first term
r=common ratio
Example:
10, 30, 90, 270, 810, 2430, ...
Calculate the 10th term of this geometric sequence:
a1 = 10 (the first term) r = 3 (the "common ratio") n = 10 (position of the term in the sequence)
Example: 10, 30, 90, 270, 810,
2430, ... 𝑎𝑛 = 𝑎1 • 𝑟𝑛−1 𝑎10 = 10 • 310−1
𝑎10 = 10 • 19683
𝑎10 = 196830
Example:
4, 2, 1, 0.5, 0.25, ...
Calculate the 6h term of this geometric sequence:
a1 = 4 (the first term) r = 1/2 (the "common ratio") n = 6 (position of the term in the sequence)
Example: 4, 2, 1, 0.5, 0.25, ... 𝑎𝑛 = 𝑎1 • 𝑟𝑛−1
𝑎6 = 4 • 1
2
6−1
𝑎6 = 4 • 0.03125
𝑎6 = 0.125
Finite Geometric Sequence: Important Formulas:
𝑆𝑛 =𝑎1 1 − 𝑟𝑛
1 − 𝑟
Summation
𝑟 =𝑎𝑛
𝑎𝑘
𝑛−𝑘
Common ratio
Important Formulas:
𝑎𝑛 = 𝑎1𝑟𝑛−1
Use to find a term in a sequence.
Finite Geometric Sequence:
Example:
3 + 6 + 12 +…+1536
Determine the sum of the geometric series:
a1 = 3 (the first term) r = 2 (the "common ratio") n = 10 (number of terms)
Example: 3 + 6 + 12 +…+1536
To find the n:
1536= 3 • 2𝑛−1 𝑎𝑛 = 𝑎1𝑟𝑛−1
1536
3=
3
32𝑛−1
1536
3=
3
32𝑛−1
512= 2𝑛−1 29 = 2𝑛−1
9=n-1 9+1=n 10=n
Example: 3 + 6 + 12 +…+1536
𝑆𝑛 =𝑎1 1 − 𝑟𝑛
1 − 𝑟
𝑆10 =3 1 − 210
1 − 2
𝑆10 =−3069
−1
Example: 3 + 6 + 12 +…+1536
𝑆10 =−3069
−1
𝑆10 = 3069
Example: Given a geometric series with
T1=−4 and T4=32. Determine the value of r.
a1= 4 a4= 32 r = ??
Example: To find for r:
𝑟 =𝑎𝑛
𝑎𝑘
𝑛−𝑘
n=position of an in the sequence k= position of ak in the sequence an=a given term in the sequence ak=a given term in the sequence
𝑟 =32
−4
4−1
𝑟 = −83
𝑟 = −2
Important Formulas:
𝑆∞ =𝑎1
1 − 𝑟
Summation
𝑟 =1 − 𝑎1
𝑆∞
Common ratio
Infinite Geometric Sequence:
Important Formulas:
𝑎1 = 𝑆𝑛(1 − 𝑟)
Use to find the first term of the sequence
Infinite Geometric Sequence:
Example:
1/2, 1/4, 1/8, 1/16, ...
Add up ALL the terms of the Geometric Sequence that halves each time:
Example: Add up ALL the terms of the Geometric
Sequence that halves each time:
𝑆∞ =
12
1 −12
𝑆∞ =𝑎1
1 − 𝑟
𝑆∞ = 1
Example:
0.9 + 0.09 + 0.009 …
Calculate the sum of this infinite geometric sequence:
Example: Add up ALL the terms of the Geometric
Sequence that halves each time:
𝑆∞ =0.9
1 − 0.1 𝑆∞ =
𝑎1
1 − 𝑟
𝑆∞ = 1
1. What is the eleventh term of the geometric sequence
3, 6, 12, 24, ... ?
(Move on to the next slide if you are done answering the problem)
What is the eleventh term of the geometric sequence 3, 6, 12, 24, ... ?
𝑎𝑛 = 𝑎1 • 𝑟𝑛−1
𝑎11 = 3 • 211−1
𝑎11 = 3 • 1024
𝑎11 = 3072
2. What is the ninth term of the
geometric sequence 81, 27, 9, 3, ... ?
(Move on to the next slide if you are done answering the problem)
What is the ninth term of the geometric sequence 81, 27, 9, 3, ... ?
𝑎𝑛 = 𝑎1 • 𝑟𝑛−1
𝑎9 = 81 • 1
3
9−1
What is the ninth term of the geometric sequence 81, 27, 9, 3, ... ?
𝑎9 = 81 •1
6561
𝑎9 =1
81
3. What is the sum of the first nine terms of the geometric
sequence 20, 10, 5, ... ? Give your answer as a
decimal correct to 1 decimal place.
(Move on to the next slide if you are done answering the problem)
What is the sum of the first nine terms of the geometric sequence 20,
10, 5, ... ?
𝑆𝑛 =𝑎1 1 − 𝑟𝑛
1 − 𝑟
𝑆9 =
20 1 −12
9
1 −12
What is the sum of the first nine terms of the geometric sequence 20,
10, 5, ... ?
𝑆9 =19.9609375
0.5
𝑆9 =39.9
4. The first term of a geometric sequence is 5
and the sixth term is 160. What is the common
ratio?
(Move on to the next slide if you are done answering the problem)
The first term of a geometric sequence is 5 and the sixth term is 160. What is the common ratio?
𝑟 =𝑎𝑛
𝑎𝑘
𝑛−𝑘 𝑟 =
160
5
6−1
𝑟 = 325
𝑟 = 2
5. Add up all the terms of the following infinite geometric sequence:
(1/3, -1/9, 1/27, -1/81,…)
(Move on to the next slide if you are done answering the problem)
Add up all the terms of the following infinite geometric sequence:
(1/3, -1/9, 1/27, -1/81,…)
𝑆∞ =𝑎1
1 − 𝑟
𝑆∞ =
13
1 − −13
Add up all the terms of the following infinite geometric sequence:
(1/3, -1/9, 1/27, -1/81,…)
𝑆∞ =1
4
:Alilam, Daryne Hannah Z. Arubio, Queencess Arrainne Berganio, Chelsea Anne D. Candelario, Krystal Mae De Guzman, Keith Mia P. 10 – Galilei Caloocan National Science and Technology High School School Year 2015 - 2016
