Summary Infinity Unveiled: IEB Grade 12 Sequences and Series
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Course
Mathematics
Institution
12th Grade
Ace your matric year! Dive into the enchanting world of sequences and series with our meticulously crafted digital notes designed exclusively for IEB Grade 12 students. Say goodbye to dull, monotonous study materials and embrace the vibrancy of learning with our fun, aesthetic, and colourful notes,...
key objectives :
Steps to solve :
key terms :
1. find values of variables .
1 Arithmetic Linear Consecutive
·
:
immediately
2 . Use these values to find ·
find the difference after each other
the general term Substitute known
· ·
common difference :
3 . Use the general term to values d = Tn + 1 -
In
find specific term values ·
find
any unknowns
·
General term: nth term
↳ Use
.
specific values to ·
Write in standard Tn = dn + C
↳ difference
find the term number . Quadratic
2
term
s ↑
. If asked
5 to find the
·
find the second Tn =
a + (n i)d -
↳ first term
sequence supply at ,
difference
↑ quadratic
Least three terms ·
2a = second difference Tn = an2 + bn + C
·
3a + b = first difference *
2a = second difference
·
a + b+ c =
first term *
sa + b = first difference
Example #1 : * a+ b + c =
first term
mi 2m + 2 ; 5m + 3 are consecutive terms of an arithmetic ·
Amount of terms : subscript
. Determine
sequence m and find the sequence and nth term of 1
, eg .
T,
00
Tz -
11 =
T3 -
72 Example #2
:
2m + 2 -m = 5m +
3-2m- 2 find the arithmetic sequence Example #3 :
-
2m = -
1 If the 6th term is 10 and the Determine the nth term of :
m = 14th term is 58 a + b +C
"
.
E ; 3: 2 Tn = a+ (n 1) d
- 6 17 34 57
17 6 17
34 57-34
-
52
-
2(z) +
2 -
(2) = 10 =
a + (b -
1)d
3a + b 11 11"
..
Tn =
a + (n 1) d
- 10 =
a + 5d -
23 -
1723
-n =
2 +
(n -
1)5z a
=
10 -
50 ... 1)
2a 6 b
5
in =
2n
-
2 58 = a + (14 1)d -
... 2)
I
Sub a = 10-5d into 2 )
.
L : In = 20 + (n 1)6 -
Tn = an + bn + C
T
58 = 10 -
5d +
13d : -
20: -
14 ;
-
8 2a = 6 3a + b = 11
24 =
4d Note to highlight what a = 3 3(3) + b =
11
d b final need b +c b b 2
your answers a
= =
+ =
:
a = 10 -
5(b) to contain and return to 3 +2 + c =
b :. Tn =
3nz + 2n + 1
a = -
20 that before moving on C =
1
, 1)
n -
1
. Geometric
3 Sequences n
=
a . r formed by multiplication
-n +
↳
J by
·
find the ratio r N = Tn a common ratio"r"
first term
·
The first term is a
·
substitute
Example #4 :
Given the sequence 6 ; 18 : 54 % continuesdetermine the next two terms the nth ,
>
-
Note :
term and how
many terms the sequence has Check the
b 18 18
54
54 118098 =
6 .
3"-1 given values
6 18
39 zn
1
-
=
against all
3 3
:. 9 =
n -
1
pattern types
n 1
Tn check for
-
= a .
+ r =
3 n =
18 le :
first to second
-1
Tn =
6 .
37 G = 6 :
10 terms differences B
:
162 ; 486 are the next two terms a ratio
Example #5 : Example #6 :
g-2 : are the first three terms The 6th term of geometric
g +; g-3 a
sequence is
3
of a geometric sequence. Determine the and the 11th term is . Determine the
27 sequence
q (3) n
-
1
1
value of 9 E E and show the first three rn Tn
-
Tn a
.
= =
.
6 1
: is
-
terms are 18 : 6 : 2 ,
hence find the nth term
.
,
3 =
a . o
Sn=Lat In-da
·
<n =
a (n 1 d Sn a (r"-1) +
·
·
=
+ -
; r
·
Tn = arn-1 r -
1
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