Grade 12
Mathematics
Topic
Sequences
& Series
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Lesson Description:
In this lesson we will deal with:
 Identifying the different types of number patterns
 Looking at the formulae needed to generate the general term of a
sequence
 Using formulae to different calculations relating to sequences
 Looking at Sigma notation
 Calculating the sum of a series
 Identifying converging number patterns
Key Concepts:
 Working with arithmetic (linear), geometric (exponential) & quadratic
series
 How to find the nth term of a quadratic sequence (constant second
difference)
 Finding a term of an arithmetic (linear) sequence
 Finding a term of a geometric (exponential) sequence
 Using simultaneous equations to find a sequence
 Develop a formula for the sum of an arithmetic and geometric series
 Working with arithmetic and geometric series
 How many terms are added to get a particular sum
 How to find a term if you are given a formula for the sum
 Convergence
 The sum to infinity
 Problems using sequences and series
 Diagrammatic problems
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Terminology & Definitions
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Sequence – a list of terms, i.e. T1; T2; T3; ...
Series – the sum of a sequence, i.e. T1 + T2 + T3 + ...
Arithmetic (Linear) Sequence – has a common first difference
Quadratic Sequence – has a common second difference
Geometric (Exponential) Sequence – has a common ratio
Constant – remains the same
Convergence – tending towards a particular value
Divergence – will continue to positive or negative infinity
Numerical – number value
Specific – a particular value
Infinite – never ending
Midpoints – half way on a line (in the middle)
Symbols, Units & Equations (Formulae)
 Although the formulae are given to you on the formulae sheet, you should
still learn them!
 The formulae sheet should only be used if you forget a formula!
 Code when you need to
 Ensure you use the correct formula!
n – number of terms in the sequence/series
Tn – the nth term of a sequence/series
Sn – the sum to n terms of a series
d – common difference of an arithmetic sequence/series
r – common ratio of a geometric sequences/series
a – first term (T1) of a sequence/series
l – last term (Tn) of a sequence/series
S – sum to infinity of a converging geometric sequence
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Arithmetic (Linear) Sequences & Series
𝑇𝑛 = 𝑎 + (𝑛 − 1)𝑑
𝑑 = 𝑇𝑛+1 − 𝑇𝑛
𝑛
[2𝑎 + (𝑛 − 1)𝑑]
2
𝑛
𝑆𝑛 = [𝑎 + 𝑙]
2
𝑆𝑛 =
Geometric (Exponential) Sequences & Series
𝑇𝑛 = 𝑎. 𝑟 𝑛−1
𝑟=
𝑇𝑛+1
𝑇𝑛
𝑎(𝑟 𝑛 − 1)
𝑆𝑛 =
;𝑟 ≠ 1
𝑟−1
Quadratic Sequences
𝑇𝑛 = 𝑎𝑛2 + 𝑏𝑛 + 𝑐
𝑎=
1
(𝑠𝑒𝑐𝑜𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒)
2
Converging Geometric Series
𝑆∞ =
𝑎
; −1 < 𝑟 < 1
1−𝑟
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Past Exam Paper Questions:
These questions have been taken from past Department of
Basic Education final Grade 12 NSC Examinations.
November 2008 Paper 1
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November 2011 Paper 1
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All past Grade 12 NSC examination papers and their memos
can be downloaded from www.education.gov.za
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