Sequences and series

Sequences and Series

1. Recurrence Relations
A recurrence relation is a function of lower/earlier terms
i.e. un+1 = aun + bun-1 +……+ λu0
A first order recurrence relation is where un+1 is given as
a function of un.
A first order linear recurrence relation as the form

un+1 = a un + b

2. Sequences/Series
A sequence is an ordered list of numbers (terms).
The first term is usually denoted by u1 and the nthterm
of a sequence is usually denoted un. We can define a
sequence by a recurrence relation.
A finite sequence is one which has a last term.
An infinite sequence is one that continues indefinitely.
A series (or progression) is the sum of the terms in an
infinite sequence.
Note a sequence can also be defined by a function
[un=f(n)]

3. Sigma Notation

෍

𝑘=𝑎

𝑏

𝑓 𝑘 = 𝑓 𝑎 + 𝑓 𝑎 + 1 + 𝑓 𝑎 + 2 + ⋯ + 𝑓(𝑏)

Example 1

෍

𝑘=1

3

(2𝑘 + 1)

Example 2

෍

𝑘=0

4

(−2)𝑘

Example 3

෍

𝑘=2

5

(𝑘2− 1)(3𝑘 + 2)

Partial Fractions can be used to find the sum of certain
series.

Example
Given that

1

4𝑘2− 2=
1

2(2𝑘 − 1)−
1

2(2𝑘 + 1)

show that

෍

𝑘=1

𝑛
1

4𝑘2− 2 = 1

2 −
1

2(2𝑛 + 1)

and hence evaluate

෍

𝑘=1

∞
1

4𝑘2− 2

Algebraic Identities can be used to find the sum of a
series.
Example
a) Show that 1

3𝑘 𝑘 + 1

𝑘 + 2 −1

3k − 1 k k + 1 = k(k + 1)

b) Deduce that

෍

𝑘=1

𝑛

𝑘 𝑘 + 1 = 𝑛(𝑛 + 1)(𝑛 + 2)

3

c) Evaluate the sum of the series

(1x2) + (2x3) + (3x4) +…+ (99x100)

Use of Standard Formulae to sum series
First note the following two properties of Σ
Property 1

෍

𝑘=1

𝑛

𝑓 𝑘 + 𝑔 𝑘

= ෍

𝑘=1

𝑛

𝑓 𝑘 + ෍

𝑘=1

𝑛

𝑔(𝑘)

Proof

෍

𝑘=1

𝑛

{𝑓 𝑘 + 𝑔 𝑘 }

={f(1) + g(1)} + {f(2) + g(2)} + … + {f(n) + g(n)}
= {f(1) + f(2) + … + f(n)} + {g(1) + g(2) + … + g(n)}

= ෍

𝑘=1

𝑛

𝑓(𝑘) + ෍

𝑘=1

𝑛

𝑔(𝑘)

Property 2
If a is a constant, then

෍

𝑘=1

𝑛

𝑎𝑓 𝑘 = 𝑎 ෍

𝑘=1

𝑛

𝑓(𝑘)

Proof

෍

𝑘=1

𝑛

𝑎𝑓 𝑘

=af(1) + af(2) + … + af(n)
=a(f(1) + f(2) + … f(n))

= 𝑎 ෍

𝑘=1

𝑛

𝑓(𝑘)

In particular, if a and b are constants

෍

𝑘=1

𝑛

𝑎𝑓 𝑘 + 𝑏𝑔 𝑘

= ෍

𝑘=1

𝑛

𝑎𝑓 𝑘 + ෍

𝑘=1

𝑛

𝑏𝑔(𝑘)

= 𝑎 ෍

𝑘=1

𝑛

𝑓 𝑘 + 𝑏 ෍

𝑘=1

𝑛

𝑔(𝑘)

Some important series
a)

෍

𝑟=1

𝑛

𝑓 𝑟 = 𝑓 1 + 𝑓 2 + 𝑓 3 + ⋯ + 𝑓(𝑛)

b)

෍

𝑟=1

𝑛

1 = 1 + 1 + 1 + 1 + ⋯ + 1 = 𝑛

c)

෍

𝑟=1

𝑛

𝑎𝑟 + 𝑏 = 𝑎 ෍

𝑟=1

𝑛

𝑟 + 𝑏 ෍

𝑟=1

𝑛

1 = 𝑎 ෍

𝑟=1

𝑛

𝑟 + 𝑏𝑛

Learn these standard formulae

෍

𝑘=1

𝑛

𝑘 = 𝑛(𝑛 + 1)

2
෍

𝑘=1

𝑛

𝑘2= 𝑛(𝑛 + 1)(2𝑛 + 1)

6
෍

𝑘=1

𝑛

𝑘3= 𝑛2(𝑛 + 1)2

4

Notice that

෍

𝑘=1

𝑛

𝑘3=

෍

𝑘=1

𝑛

𝑘

2

where a
and b are
constants

Example 1
Evaluate

෍

𝑘=1

10

(2𝑘3− 3𝑘)

Example 2
Evaluate the sum of the series
(1x42) + (2x52) + (3x62) + …+ (15x182)

Example 3
Obtain an expression in terms of n for

෍

𝑘=1

𝑛

(2𝑘2− 5)

Give your answer as a single algebraic fraction in it’s
simplest from.

Example 4
Evaluate

෍

𝑘=25

45

𝑘2

Example 5
a) Obtain an expression for

෍

𝑘=1

𝑛

𝑘(𝑘 + 1)(2𝑘 − 1)

in terms of n. Give your answer as a single fraction in it’s
simplest form.
b) Hence evaluate the sum of the series
(1x2x1) + (2x3x3) + (3x4x5) + (4x5x7) + … + (20x21x39)

Arithmetic Sequence and Series

A sequence is termed arithmetic if there is a common
difference between consecutive terms.
i.e. u1, u2, u3… is arithmetic if
u2 – u1 = u3 – u2 = … = un+1 – un = d where d is a constant.
We usually denote u1 by a.
In general, an arithmetic sequence has terms
u1 = a
u2 = u1 + d = a + d
u3 = u2 + d = a + 2d
u4 = u3 + d = a + 3d
i.e. the nthterm un = a + (n - 1)d
𝑛 ∈ 𝑵

An arithmetic series is a sum of the terms of an
arithmetic sequence. The sum of the first n terms
(“sum to n terms”) is calculated using the formula

𝑆𝑛 = 𝑛

2 [2𝑎 + 𝑛 − 1 𝑑]

Example 1
Find the 20thterm of the arithmetic sequence 3, 7, 11,…

Example 2
Find the nthterm of the arithmetic sequence 21, 15, 9, 3,…

Example 3
The 3rdterm of an arithmetic sequence is 10 and the 22nd
term is 67.
a) Find the first term and the common difference.
b) Find the 17thterm of the sequence.

Example 4
Given the arithmetic sequence 2, 8, 14, 20,… which term
is the first to exceed 65?

Example 5
Find the sum of the arithmetic series 4 + 7 + 10 + 13 + …
to 10 terms.

Example 6
Calculate the sum of the arithmetic series 92 + 90 + 88
+ 86 + … + 48

Geometric Sequence and Series
A sequence is terms geometric if there is a common ratio
between consecutive terms.
i.e. u1, u2, u3, u4,… is geometric if 𝑢𝑛

𝑢𝑛+1= 𝑟 where r is a

constant.
In general a geometric sequence has terms
u1 = a
u2 = u1 x r = ar
u3 = u2 x r = ar2
u4 = u3 x r = ar3

un = arn-1n є N
A geometric series is a sum of the terms of a geometric
sequence.
The sum of the first n terms (“sum to n terms”) is
calculated using the formula

𝑆𝑛 =𝑎(𝑟𝑛−1)

𝑟−1
𝑟 > 1 OR 𝑆𝑛 =𝑎(1−𝑟𝑛)

1−𝑟
𝑟 < 1

Example 1
Find the 12thterm of the geometric sequence 7, 14, 28,
56…

Example 2
If the first two terms of a geometric sequence are 2
and 6, which term is 13122?

Example 3
A geometric sequence of positive terms has 12 and 192
as the third and seventh terms. Find the 10thterm of
the sequence.

Example 4
Find the sum to 13 terms of the geometric series
-16 – 8 – 4 – 2 – 1 -…

Example 5
A geometric series has a common ratio of -3. It’s sum
to 5 terms is 122. Calculate the first term and the sum
to seven terms.

Infinite Series and Sum to Infinity
The sum of the first n terms of a series, Sn, is known as
a partial sum of the series.
When the partial sum tends towards a limit as n tends to
infinity, then the limit is called the sum to infinity of the
series and is denoted by S∞
Note: S∞ for an arithmetic sequence is undefined!
The sum of the first n terms is

𝑆𝑛 = 𝑎(1 − 𝑟𝑛)

1 − 𝑟

if -1 < r < 1 then rn-> 0 as n -> ∞ and

𝑆𝑛 = 𝑎(1 − 0)

1 − 𝑟
=
𝑎

1 − 𝑟

For a geometric sequence there are 2 cases to consider
1. |r| > 1, S∞ is undefined
2. |r| < 1, S∞ exists and S∞ = 𝑎

1−𝑟

Example 1
Does a sum to infinity exist? If yes, find it.
a) 4 + 1 + 1

4+ …

b) 0.2 – 0.6 + 1.8 - …

Example 2
u1, u2, u3,… is a geometric sequence with u1 = 48 and

෍

𝑘=1

∞

𝑢𝑘 = 64

Find the value of u4.

Expanding (a + b)-1
We have seen before that we can expand (x + y)nusing
the binomial theorem.

(𝑥 + 𝑦)𝑛= ෍

𝑟=0

𝑛

𝑛
𝑟 𝑥𝑛−𝑟𝑦𝑟,
𝑛
𝑟
=
𝑛!

𝑛 − 𝑟 ! 𝑟!

However before we stated that n є N. Now we want to
consider n = -1.
The geometric series 1 + x + x2+ x3+ …

|x| < 1

has a sum to infinity of 1

1−𝑥= (1 − 𝑥)−1

i.e.

(1 − 𝑥)−1= ෍

𝑟=0

𝑛

𝑥𝑟= 1 + 𝑥 + 𝑥2+ 𝑥3+ ⋯

Using the binomial theorem we get

(1 − 𝑥)−1

= −1

0
1−1(−𝑥)0+
−1
1
1−2(−𝑥)1+
−1
2
1−3(−𝑥)2+ ⋯

=
−1 !

−1 ! 0 ! +
−1 !
−2 ! 1! −𝑥 +
−1 !

−3 ! 2! 𝑥2 + ⋯

= 1 + −1

−𝑥 +(−1)(−2)

2
𝑥2+ ⋯

= 1 + 𝑥 + 𝑥2+ ⋯

i.e. a geometric series with first term 1 and a common
ratio x.
We can extend this idea to (a + b)-1

(𝑎 + 𝑏)−1=
1

𝑎 + 𝑏 =
1

𝑎(1 + 𝑏

𝑎)

=

1
𝑎

1 − (− 𝑏

𝑎)

i.e. (a + b)-1generates a geometric series with first term
1
𝑎and common ratio − 𝑏

𝑎

Note: The derivative of the series tends to the
derivative of the limit.
The integral of the series tends to the integral of the
limit.

𝑎

1 − 𝑟

Example 1
Expand (3 – x)-1

Example 2
Expand (2 + 3x)-1

Example 3
Expand

1
0.9to 4 decimal places