Sequences and Series
Terminology
Sequence: Numbers ordered according to a specific rule.
Term: Each number in the sequence.
n: Position of the term.
Tn : Value of the term.
General term: Shows the relationship between n and Tn.
a: The first term.
d: The constant difference.
r: The constant ratio.
l: The last term.
Series: Refers to the sum of a sequence. (Sn)
Sigma: Shorthand notation to describe the sum of.( )
notation: Notation to describe the sum of a series of terms, obeying a general rule.
Number pattern revision
Well known patterns
Even numbers: 2; 4; 6; 8; 10; …
. Tn = 2n
Odd numbers: 1; 3; 5; 7; 9; …
. Tn = 2n – 1
The squares: 1; 4; 9; 16; 25; …
. Tn = n2
Triangular numbers: 1; 3; 6; 10; 15; …
𝑛 (𝑛+1)
. Tn =
2
The cubes: 1; 8; 27; 64; 125; …
. Tn = n3
Powers of 2: 2; 4; 8; 16; 32; …
. Tn = 2n
,Linear
(1st difference constant)
Tn = an + b
Common difference = a
Sub one value and it’s n into equation and solve = b
Quadratic
(2nd difference constant)
Tn = an2 + bn + c
Common difference = 2a
T0 = c
Sub in a and c, but in value and it’s n into equation and solve = b
Cubic
(3rd difference constant)
Tn = an3 + bn2 + cn + d
, Arithmetic Sequences
e.g. 2; 5; 8; 11; 14; …
Constant difference = d
d = Tn – Tn-1
. = T2 – T1
. = T3 – T2
Linear formula:
T1 = a
T2 = a + d
T3 = a + 2d
Tn = a + (n – 1) d
Type 1: Find the value of a term
Determine the 30th term of the sequence 45; 50; 55; …
Tn = a + (n – 1) d
a = 45
d=5
T30 = 45 + (30 – 1) 5
. = 190
Type 2 : Find the number of terms
How many terms are there in the sequence 13; 10; 7; …; - 44
Tn = a + (n – 1) d
a = 13
d=-3
-44 = 13 + (n – 1) - 3
. n = 20
Type 3: Find the sequence
Find the sequence if the 10th term is 31 and the 15th term is 46
31 = a + (10 – 1) d 46 = a + (15 – 1) d
31 = a + 9d 46 = a + 14d
. a = 31 – 9d . a = 46 – 14d
31 – 9d = 46 – 14d a = 31 – 9(3)
d=3 a=4
Tn = 4 + (n – 1) 3
4; 7; 10; …
Terminology
Sequence: Numbers ordered according to a specific rule.
Term: Each number in the sequence.
n: Position of the term.
Tn : Value of the term.
General term: Shows the relationship between n and Tn.
a: The first term.
d: The constant difference.
r: The constant ratio.
l: The last term.
Series: Refers to the sum of a sequence. (Sn)
Sigma: Shorthand notation to describe the sum of.( )
notation: Notation to describe the sum of a series of terms, obeying a general rule.
Number pattern revision
Well known patterns
Even numbers: 2; 4; 6; 8; 10; …
. Tn = 2n
Odd numbers: 1; 3; 5; 7; 9; …
. Tn = 2n – 1
The squares: 1; 4; 9; 16; 25; …
. Tn = n2
Triangular numbers: 1; 3; 6; 10; 15; …
𝑛 (𝑛+1)
. Tn =
2
The cubes: 1; 8; 27; 64; 125; …
. Tn = n3
Powers of 2: 2; 4; 8; 16; 32; …
. Tn = 2n
,Linear
(1st difference constant)
Tn = an + b
Common difference = a
Sub one value and it’s n into equation and solve = b
Quadratic
(2nd difference constant)
Tn = an2 + bn + c
Common difference = 2a
T0 = c
Sub in a and c, but in value and it’s n into equation and solve = b
Cubic
(3rd difference constant)
Tn = an3 + bn2 + cn + d
, Arithmetic Sequences
e.g. 2; 5; 8; 11; 14; …
Constant difference = d
d = Tn – Tn-1
. = T2 – T1
. = T3 – T2
Linear formula:
T1 = a
T2 = a + d
T3 = a + 2d
Tn = a + (n – 1) d
Type 1: Find the value of a term
Determine the 30th term of the sequence 45; 50; 55; …
Tn = a + (n – 1) d
a = 45
d=5
T30 = 45 + (30 – 1) 5
. = 190
Type 2 : Find the number of terms
How many terms are there in the sequence 13; 10; 7; …; - 44
Tn = a + (n – 1) d
a = 13
d=-3
-44 = 13 + (n – 1) - 3
. n = 20
Type 3: Find the sequence
Find the sequence if the 10th term is 31 and the 15th term is 46
31 = a + (10 – 1) d 46 = a + (15 – 1) d
31 = a + 9d 46 = a + 14d
. a = 31 – 9d . a = 46 – 14d
31 – 9d = 46 – 14d a = 31 – 9(3)
d=3 a=4
Tn = 4 + (n – 1) 3
4; 7; 10; …
