Topic 1: Sequences
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Numbers:
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N = {0, 1, 2, 3, … }
Z = { … -2, -1, 0, 1, 2, … }
Q = { a/b, a, bEZ, b≠0 }
R = Rational + Irrational
Irrational = { “pi number”, e, √2, √3, √13, √15 }
{ x: x >10, xER }
Any number is written in the form a . 10'n, where 1 < a < 10
For example: 1, 25 . 10'24
 Exp.: Find the scientific notation of the numbers:
 a) 125,982 . 10'9 = 1,25982 . 10'11
 b) 0,000057.10'12 = 0,57 . 10'7
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Exp.: Consider the numbers:
a) x = 3 . 10'7
b) y = 4 . 10'9
Find x + y and x . y in scientific form
x+y=?
b) x . y = ?
Ans.: x + y = 3 . 10'9 + 400 . 10'7 = 403 . 10'7 = 4,03 . 10'9
x . y = 3 . 10'7 . 4 . 10'9 = 1,2 . 10'16 = 1,2 . 10'17
 Sequences:
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A sequence is a list of numbers with a rule.
For example: { 2, 4, 6, 8, 10, ... } Rule: +2 / Even Numbers
{ 1, 4, 7, 10, 13, ... } Rule: +3
{ 2, 4, 8, 16, 32, ... } Rule: 2k
{ 3, 9, 27, 81, ... } Rule: 3k
 First 2 examples are "Arithmetic Sequences" and the last 2
examples are "Geometric Sequences" examples.
 The sequences showing U’n EZ'+
 Exp.: Un = 3n + 1 Find the first three terms.
 Ans.: U1 = 3.1 + 1 = 4
U2 = 3.2 + 1 = 7
U3 = 3.3 + 1 = 10
 Exp.: Find U5 if UN = 3n – 2 / n + 5.
 Ans.: 3.5 + 2 / 5 + 5 = 13 / 10 = 1,3
 Exp.: Find the general term of the sequence, 2, 7, 12, 17, … .
 Ans.: Un = 5n – 3
 Exp.: Find the general term of the sequence, 7, 11, 15, 19, … .
 Ans.: Un = 4n + 3
 Exp.: A sequence is defined by Un = 2n – 1:
a) Find the U4.
b) Simplify Un + 1 – Un.
 Ans.: a) U4 = 2.4 – 1 = 7 b) Un + 1 = 2. (n + 1) – 1 = 2n + 2 – 1 =
2n + 1
Un+1 - Un = 2n + 1 – (2n – 1) = 2n + 1 – 2n + 1 = 2
 Exp.: The sequence is defined by Un = n . 2’n
a) Find U1 and U3.
b) Find and Simplify Un + 1 / Un – 1
 Ans.: a) U1 = 1 . 2 = 2 U3 = 3 . 2’3 = 24
b) Un + 1 / Un – 1 = (n
+ 1) . 2’n+1 / (n – 1) . 2’n-1 = n + 1 / n – 1 . 2’n+1 – n+1 = n + 1 / n
– 1 . 2’2 = 4 / 1 . n + 1 / n – 1 = 4n + 4 / n -1
 Recursive Sequence Uses Previous Term or Terms to
Find the Next Term or Terms:
 For example Un + 1 = 3 . Un
 Exp.: Un = 2 . Un-1 + 5 and U1 = 3 Find U3 = ?
 Ans.: n = 2 U2 = 2 . U1 + 5 = 11 n = 3 U3 = 2 . U2 + 5 = 27
 Exp.: Un+1 = Un + Un-1 and U1 = 5, U2 = 7 Find U4 = ?
 Ans.: n = 2 U3 = U2 + U1 = 7 + 5 = 12 n = 3 U4 = U3 + U2 =
12 + 7 = 19
 Arithmetic Sequences:
 A sequence is arithmetic when the same value is added to each
term to get the next term.
 For example: 1, 4, 7, 10, 13, …
3 + 3 = 1 + 2.3 = 7
3 + 3 + 3 + 3 = 1 + 4.3
a1 = 1 a2 = 1 + 3 = 4
a4 = 1 + 3 + 3 + 3 = 1 + 3.3
a3 = 1 +
a5 = 1 +
*** Main formula of the Arithmetic Sequences: an = a1 + (n – 1)
. d ***
 For example: a3 = a2 + d a3 = a1 + 2d (1 + 2 = 3)
6d (1 + 6 = 7) a18 = a1 + 17d (1 + 17 = 18)
 Exp.: Consider the sequence 1, 7, 13, 19, … :
 a) Is it arithmetic sequence?
 b) Find the general term.
a7 = a1 +
 c) Find U50.
 d) Ux = 475 x = ?
 Ans.: a) Yes, this is an arithmetic sequence, because the numbers
are getting 6 times bigger everytime.
b) Un = 6n - 5
c)
U50 = 6.50 – 5 = 300 – 5 = 295
d) Ux = 6n – 5 = 475 Ux = 6n
= 475 + 5 = 480 Ux = 6n = 480 n = 80 x = 80
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Infinite Sequence:
 For example: 2, 4, 6, … is an infinite sequence.
 Finite Sequence:
 For example: 3, 6, 9, … , 81 is a finite sequence.
 Exp.: 35, 39, 43, … ,95 ?
 Ans.: d = 4 U1 = 35 35 + (n – 1) .4 = 95
n =16
 Exp.: If U1 = 300 d = -5 U4 = ?
 Ans.: 300 + (n-1) .5
U4 = U1 + 3d
15 = 285
35 + 4n – 4 = 95
U4 = 300 + 3. (-5) = 300 –
 Exp.: U5 = 7 U8 = 16 U = 12 = ?
 Ans.: U8 = U5 + 3d
16 = 7 + 3d
9 = 3d
4d = 16 + 12 = 28
 Exp.: U5 = 5 and d = 7 which term is 355?
 Ans.: U5 = U1 + 4d 5 = U1 + 28 U1 = 28 -5
23 + (n – 1) .7
355 = -23 + 7n – 7
385 = 7n
n = 55
d=3
U12 = U8 +
U1 = -23
Un = -
 Exp.: The first four terms of arithmetic sequence are 2, a – b, 2a + b
+ 7 and a – 3b, (where a, b constant). Find a and b = ?
 Ans.: U1 = 2
U2 = a – b
U3 = 2a + b + 7
U4 = a – 3b d
= a2 – a1 = a3 – a2 = a4 -a3
a – b – 2 = a + 2b + 7 = -4b – a - 7
3b = -9 b = -3 2a = 2b – 14
2a = -6 – 14 2a = -20
a=
-10
b = -3 and a = -10
 Geometric Sequence:
 A sequence is called geometric when each term is multiplied by the
same value to get the next term.