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