Geometric sequence

1. UNIT 3:
GEOMETRIC SEQUENCE
1. Definition
3. General term
2. Properties of terms
4. Sum of first n terms
Designed by Le Thi My Duc19/01/2018

2. Indian King - Seram Inventor - Seta
1 2 4 8 16 32 64 128
1 𝑠𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 → 1 grain of rice
2 𝑛𝑑 𝑠𝑞𝑢𝑎𝑟𝑒 → 2 grains of rice
3 𝑟𝑑 𝑠𝑞𝑢𝑎𝑟𝑒 → 4 grains of rice
4 𝑡ℎ 𝑠𝑞𝑢𝑎𝑟𝑒 → 8 grains of rice
64 𝑡ℎ 𝑠𝑞𝑢𝑎𝑟𝑒 → ? grains of rice
What kind of prize would you like?
My Lord, I wish…
Chessboard
?
Designed by Le Thi My Duc19/01/2018

3. Geometric sequence is a (finite or infinite) sequence, where
each term after the first is found by multiplying the preceding
term by a constant q.
(un) is a geometric sequence  n ≥ 2, un= un-1.q
Number q is called common ratio of the geometric sequence
I.Definition
UNIT 3: GEOMETRIC SEQUENCE
Designed by Le Thi My Duc19/01/2018

4. a. 1, 2, 4, 8, 16, 32, 64, 128, …
Example: Find u1 and q
u1 = 1 and q = 2
b. -3, 9, -27, 81, -243.
u1 = -3 and q = -3
It’s a geometric sequence with
It’s a geometric sequence with
Designed by Le Thi My Duc19/01/2018

5. - For q = 0, a GS is of the form u1, 0, 0, …, 0,…
- For u1 = 0, a GS is of the form 0, 0, 0, …, 0,…
- For q = 1, a GS is of the form u1, u1, u1, …, u1,…
Notes
Designed by Le Thi My Duc19/01/2018

6. In a geometric sequence, the square of each term (except the
first and last terms) is the product of its consecutive terms.
II. Properties of terms
Theorem 1
𝒖 𝒌
𝟐
= 𝒖 𝒌−𝟏.𝒖 𝒌+𝟏 ; k ≥ 2
Designed by Le Thi My Duc19/01/2018

7. = −9999 < 0
= −99 . 101
Solution
𝑢100
2
= 𝑢99 . 𝑢101
Example: Is there any geometric sequence 𝑢 𝑛
including 𝑢99 = −99 and 𝑢101 = 101?
(nonsense)
Designed by Le Thi My Duc19/01/2018

8. If a geometric sequence has the first term u1 and common ratio
q ≠ 0, then the general term is defined by the formula
III. General term
Theorem 2
𝒖 𝒏 = 𝒖 𝟏. 𝒒 𝒏−𝟏
; 𝒏 ≥ 𝟐
Designed by Le Thi My Duc19/01/2018

9. 𝑢10 = −1024Answer
Example: Given a geometric sequence 2, −4 , 8 , −16 , …
Find 𝑢10.
Designed by Le Thi My Duc19/01/2018

10. Seram Seta
1 2 4 8 16 32 64 128
1 𝑠𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 → 1 grain
2 𝑛𝑑 𝑠𝑞𝑢𝑎𝑟𝑒 → 2 grains
3 𝑟𝑑 𝑠𝑞𝑢𝑎𝑟𝑒 → 4 grains
4 𝑡ℎ 𝑠𝑞𝑢𝑎𝑟𝑒 → 8 grains
64 𝑡ℎ 𝑠𝑞𝑢𝑎𝑟𝑒 → ? grains
Chessboard
?
𝑢64 = = 1. 263
= 263
Thus: The number of grains of rice on
the 64th square is
𝑢1. 𝑞63
263 = 9,223372037.1018263
Designed by Le Thi My Duc19/01/2018

11. Given geometric sequence (un) with common ratio q
IV. Sum of first n terms
 n
1
n
u 1 q
S
1 q



n 1S n.u
* q ≠ 1:
* q =1 :
Designed by Le Thi My Duc19/01/2018

12. 1 2 4 8 16 32 64 128
Seram Seta
64
64 1
1 q
S = u .
1 q


 64
1 1 2
1 2



64
2 1 
Chessboard
18.446.744.073.709.551.615
263
Designed by Le Thi My Duc19/01/2018

13. E.coli
After 10 divisions, how many new cells will be formed in total ?
F
F1
F1.1 F1.2
F2
F2.1 F2.2
20 mins
… … … …
20 mins 20 mins
Designed by Le Thi My Duc
After 10 divisions, the number of cells is
u11 = u1.q10 = 1.210 = 1024 cells
19/01/2018

14. Paper Sizes
Georg Lichtenberg
A6
A5
A4
A3
A2
A1
A0
A6
 In 1786, Georg Lichtenberg proposed to standardise paper size
 In the 1922 Germany began the standardisation with the
introduction of “Deutsche Industrie Normen” (D.I.N.). DIN was
subsequently replaced by “International Organization for
Standardization” (I.S.O)
A1 A2 A3 A4 A5 …
841 594 420 297 210
594 420 297 210 148
Length
(mm)
Width
(mm)
 The advantages of standardization are obvious for efficiency of paper
production, print production, storage, etc. hence “International Paper
Sizes” (I.P.S.) was born.
 Produced paper sizes based on 3 series A, B & C. “A” series are the
most popular and most practical
…
…
Designed by Le Thi My Duc19/01/2018

15. Definition
Properties
General term
Sum of
first n terms
𝑢 𝑛 = __________ ; 𝑛 ≥ 2
 n
1u 1 q
1 q


1n.u
𝑢 𝑘
2
= ___________ ; k ≥ 2
(un) is a _________ ________
 n ≥ 2, ________
geometric sequence
un= un-1.q
𝒖 𝒌−𝟏.𝒖 𝒌+𝟏
𝒖 𝟏. 𝒒 𝒏−𝟏
nIf q 1, S 
nIf q 1, S 
Designed by Le Thi My Duc19/01/2018

16. Designed by Le Thi My Duc
Question 1: Given a geometric sequence (un) with u1 = 2, u6 = 486.
Find q.
Question 2: What is the eleventh term of the geometric sequence
3, 6, 12, 24, … ?
A. 66 D. 3072B. 786 C. 1536
Question 3: Given a geometric sequence(un) with u1 = 3, common
ratio q = 2. What ordinal of the term is number 192?
A. 10 D. 7B. 9 C. 8
Question 4: In a geometric sequence (un). What is the formula of Sn
with q ≠ 1?
Question 5: Given a geometric sequence (un) with u1 = 2, q = -0.5.
Find u9.
19/01/2018