Sequences and Series

SEQUENCES

 An ordered list of numbers is called a sequence.
 There is a rule to specify how the sequence continues.
 The numbers in a sequence are called the terms of the sequence.
 The terms of a sequence are referred to as 𝑻𝟏 ; 𝑻𝟐 ; 𝑻𝟑 … 𝑻𝒏 , where 𝒏 is the position of a term
in the sequence.
 Where 𝑻𝒏 is the last term in a sequence, 𝒏 is the number of terms in the sequence.

You have previously identified a variety of patterns or sequences: linear, quadratic, exponential, perfect
squares or cubes, and others.

In this topic we look more closely at
1) Linear sequences which we now refer to as Arithmetic.
2) Patterns or sequences in which the variable is in the exponent, which we now refer to as
Geometric.

1. ARITHMETIC SEQUENCES (or Arithmetic Progressions AP’s)

 A sequence is Arithmetic if each successive term is formed by adding the same value to the
preceeding term (i.e. a linear pattern).
 The value that is added is called 𝑑, the constant difference.

e.g. 4 ; 8 ; 12 … is arithmetic because you form the sequence by adding 4 each time to
form the next term, therefore 𝑑 = 4.

e.g. 20 ; 15 ; 10 … is arithmetic because you form each new term by adding
−5 to the previous term, therefore 𝑑 = −5.

NB You always think of adding the constant difference, rather than subtracting.

The Test for an Arithmetic Sequence
Given three terms 𝑇1 ; 𝑇2 and 𝑇3 ,
if 𝑇3 − 𝑇2 = 𝑇2 − 𝑇1 , then the sequence is arithmetic (because the difference between each
successive term is constant).

e.g. Is the sequence 110 ; 117 ; 123 … arithmetic?

Ans:




1

, e.g. The terms 𝑥 − 2 ; 2𝑥 ; 𝑥 + 4 form an arithmetic sequence. Find 𝑥.

Ans:




Do Pg 2 Ex 1.1 # 1
And # 2 and 4
(these require you to remember quadratic patterns from last year.)


The GENERAL TERM of an ARITHMETIC SEQUENCE

A term in an arithmetic sequence can be expressed according to an algebbraic rule.
Calling the first term 𝑎, and the constant difference 𝑑, we see that:
𝑇1 = 𝑎
𝑇2 = 𝑎 + 𝑑
𝑇3 = 𝑎 + 2𝑑
𝑇4 = 𝑎 + 3𝑑 etc.
Each term has ‘one less than the term
Do you see the pattern? number’ of 𝑑′𝑠 added on to 𝑎.

Hence all arithmetic sequences can be written in general algebraic terms as
𝑎 ; 𝑎 + 𝑑 ; 𝑎 + 2𝑑; … ; 𝑎 + (𝑛– 1)𝑑.
Note that the last term, 𝑇𝑛 , has ‘one less
than 𝑛’ number of 𝑑′𝑠 added on to 𝑎.
The 𝑛 𝑡ℎ term (known as the general term) defines the rule of the sequence. This rule binds together
the values of 𝑎, 𝑑, 𝑛 and 𝑇𝑛 .

The GENERAL TERM of an Arithmetic Sequence is 𝑻𝒏 = 𝒂 + (𝒏 – 𝟏)𝒅

and the formula 𝑻𝒏+𝟏 − 𝑻𝒏 = 𝑻𝒏 − 𝑻𝒏−𝟏 is true.


NOTE: Only the FIRST formula appears on the Gr 12 formula sheet.

e.g. Determine the eleventh term of the arithmetic sequence 5 ; 11 ; 17 …

Ans: In general, once we know that a
sequence has an arithmetic
pattern (is an AP), write down the
general term , sub in, and solve for
the unknown variable.


2

, e.g. Which term of the arithmetic sequence 25 ; 20 ; 15 … is equal to −20?

Ans:




e.g. The first term of an arithmetic sequence is 10 and the sixth term is 85.
Find the constant difference.

Ans:




e.g. The tenth term of an arithmetic sequence with constant difference −2 is 80. Determine the
first term.

Ans:




Do Pg 4 Ex 1.2 # 1 c) d) g) h) i)

Do Pg 7 and 8 Ex 1.3 # 1 LHC (left hand column)
# 2 (i) (ii) (iii) for LHC
# 3 a) c) e)
# 9 all



3