DP IB Analysis &
Approaches (AA): HL
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Sequences & Series
Contents
Language of Sequences & Series
Sigma Notation
Arithmetic Sequences & Series
Geometric Sequences & Series
Applications of Sequences & Series
Compound Interest & Depreciation
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Language of Sequences & Series
Language of sequences & series
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What is a sequence?
A sequence is an ordered set of numbers with a rule for finding all of the numbers in the
sequence
For example 1, 3, 5, 7, 9, … is a sequence with the rule ‘start at one and add two to
each number’
The numbers in a sequence are often called terms
The terms of a sequence are often referred to by the letter u with a subscript
So in the sequence above, u
The nth term is u
1
= 1, u 2 = 3, u 3 = 5, ...
n
If a formula for the nth term is given, then each term in the sequence can be found by
substituting the term number, n , into the formula
e.g. if u
n
= 2n − 1then the 5th term is u 5 = 2 × 5 − 1 = 9
What is a series?
You get a series by summing up the terms in a sequence
E.g. For the sequence 1, 3, 5, 7, … the associated series is 1 + 3 + 5 + 7 + …
The notation S is used to refer to the sum of the first n terms in a series
n
S n = u 1 + u 2 + u 3 + ... u n
For the series above, S
4
= 1 + 3 + 5 + 7 = 16
Worked Example
Determine the first five terms and the value of S in the sequence given by
u n = 5 − 2n .
5
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Sigma Notation
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Sigma notation
What is sigma notation?
Sigma notation shows the sum of a certain number of terms in a sequence
The symbol Σ is the capital Greek letter 'sigma' and stands for 'sum'
∑= r
4
e.g. for
2
r 1
The formula after the Σ tells you how to work out each term
It is the nth term formula but written in r instead
The lower limit tells you which term to start on
The upper limit tells you which term to end on
∑= r = 1 + 2 + 3 + 4 = 30
4
So
2
2
2
2
2
r 1
The letter k can also be used (instead of r )
Be careful, as not all lower limits start at 1
∑
4
For example
k =0
∑ 2k − 13
14
k 3 or
k =7
(
)
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Examiner Tips and Tricks
Your GDC can use sigma notation, which gives you a good way to check your answers
to questions involving summing terms!
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Worked Example
A sequence is defined by u
n
= 2 × 3n −1 for n ∈ℤ+ .
(a) Write an expression for u
1
+ u 2 + u 3 + ... + u 6 using sigma notation.
(b) Write an expression for u
7
+ u 8 + u 9 + ... + u 12 using sigma notation.
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Arithmetic Sequences & Series
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Arithmetic sequences
What is an arithmetic sequence?
An arithmetic sequence is a sequence in which the difference between consecutive
terms is constant
This constant difference is known as the common difference, d
e.g. for the sequence 1, 4, 7, 10, … d
=3
A negative common difference means the sequence decreases
How do I find a term in an arithmetic sequence?
The nth term formula for an arithmetic sequence is
un = u1 + n − 1 d
(
)
Where u is the first term, and d is the common difference
1
Examiner Tips and Tricks
The formula for the nth term in an arithmetic sequence is given in the formula booklet.
Examiner Tips and Tricks
In harder arithmetic sequence questions, it is common to have to solve simultaneous
equations in u and d .
1
Worked Example
The fourth term of an arithmetic sequence is 10 and the ninth term is 25.
Find the first term and the common difference of the sequence.
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Arithmetic series
How do I find the sum of an arithmetic sequence?
The formulae to find the sum of the first n terms of an arithmetic sequence, S , are
n
n
2u 1 + n − 1 d
2
n
Sn = u 1 + u n
2
Sn =
(
(
(
)
)
)
Where
u 1 is the first term
d is the common difference
u n is the last term
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Examiner Tips and Tricks
Both formulae for the sum of the first n terms in an arithmetic sequence are given in
the formula booklet (use which ever one is more convenient for the question).
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Examiner Tips and Tricks
In harder questions on arithmetic series, it is common to have to solve quadratics in n
or simultaneous equations in u and d .
1
Worked Example
The sum of the first 10 terms of an arithmetic sequence is 630.
(a) If the first term is 18, find the common difference, d .
(b) If, instead, the common difference is 11, find the first term of the sequence.
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Geometric Sequences & Series
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Geometric sequences
What is a geometric sequence?
A geometric sequence is a sequence where you multiply by a fixed number each time to
get the next term in the sequence
The fixed number that you multiply by is called the common ratio, r
e.g. for 2, 6, 18, 54, 162, … r
=3
A geometric sequence can be
>1
decreasing if 0 < r < 1
alternating if r < 0
increasing if r
changing between positive and negative
How do I find a term in a geometric sequence?
The nth term formula for a geometric sequence is
u n = u 1rn −1
Where u is the first term and r is the common ratio
1
Examiner Tips and Tricks
The formula for the nth term of a geometric sequence is given in the formula booklet.
Examiner Tips and Tricks
Some good tricks for geometric sequences are
dividing two terms by each other to eliminate u when finding r
1
using logarithms when finding n
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Worked Example
The sixth term of a geometric sequence is 486 and the seventh term is 1458.
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Find
(a) the common ratio of the sequence,
(b) the first term of the sequence.
Geometric series
How do I find the sum of a geometric sequence?
The formulae for the sum of the first n terms of a geometric sequence are:
Sn =
u 1 rn − 1
(
r−1
)
=
u 1 1 − rn
(
)
1−r
where
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u 1 is the first term
Your notes
r is the common ratio
Examiner Tips and Tricks
Both formulae for the sum of the first n terms of a geometric sequence are given in the
1 and the second when r 1 ).
formula booklet (the first one is easier when r
>
<
Examiner Tips and Tricks
Harder questions on geometric series may require the use of logarithms.
Worked Example
A geometric sequence has a first term of 25 and a common ratio of 0.8.
Find the fifth term and the sum of the first five terms.
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Sum to infinity
What is the sum to infinity of a geometric series?
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The sum to infinity of a geometric series, S , is the sum of all the terms in a geometric
∞
sequence (infinitely many terms)
> 1 or r < − 1 (written |r | > 1 )
the sum to infinity is infinity, ∞
e.g. 1 + 2 + 4 + 8 + 16 + ... → ∞
For r
This geometric series is said to diverge
− < r < 1 (written |r | < 1 )
For 1
the sum to infinity is a finite value (called a limit)
e.g. 1
1
1
1
1
+ 2 + 4 + 8 + 16 + ... → 2
The limit is 2
This geometric series is said to converge
What is the condition for convergence?
For the sum to infinity of a geometric series to converge to a finite value (a limit), the
condition required is
|r | < 1
| | < 1 means −1 < r < 1
How do I calculate the sum to infinity?
If | r | < 1 , then the sum to infinity of a convergent geometric series can be calculated
where r
by the formula
S∞ =
u1
1−r
where
u 1 is the first term
r is the common ratio
|r | < 1
The value calculated by the formula is the limit of the series
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Examiner Tips and Tricks
The formula for the sum to infinity of a geometric series is given in the formula booklet,
as well as the condition for convergence, r
1.
| |<
Worked Example
The first three terms of a geometric sequence are 6
2
, 2, 3.
Explain why the series converges and find the sum to infinity.
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Applications of Sequences & Series
Applications of arithmetic sequences &
series
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How do I spot arithmetic sequences and series?
If a quantity is changing by repeatedly having a fixed amount added to or subtracted
from it, then use arithmetic sequences and series
For example
simple interest (the same amount of interest is added to an initial amount each
year)
stacking cups
planting trees
counting dots in a pattern
Examiner Tips and Tricks
If you don't spot an arithmetic sequence in the exam, you can still work out some parts
by hand, but you won't have the u and S formulae to speed you up.
n
n
Worked Example
Jasper is saving for a new car.
He puts USD $100 into his savings account and then each month he puts in USD $10
more than the month before.
Jasper needs USD $1200 for the car.
Assuming no interest is added, find
(a) the amount Jasper has saved after four months,
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(b) the month in which Jasper reaches his goal of USD $1200.
Applications of geometric sequences &
series
How do I spot geometric sequences and series?
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If a quantity is changing by repeatedly being multiplied (or divided) by the same
amount each time, use geometric sequences and series
For example
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compound interest (interest each year is on the initial amount of money plus all
the interest you've saved up to that point)
population growth
bacterial growth
radioactive decay
Examiner Tips and Tricks
If you don't spot a geometric sequence in the exam, you can still work out some parts
by hand, but you won't have the u and S formulae to speed you up.
n
n
Worked Example
A new virus is circulating on a remote island.
On day one, there were 10 people infected. The number of new infections increases
at a rate of 40% per day.
(a) Find the expected number of people who are newly infected on the 7th day.
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(b) Find the expected number of infected people on the island after one whole week,
assuming no one has recovered.
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Compound Interest & Depreciation
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Compound interest
What is compound interest?
Compound interest is where interest is added to both:
the initial investment
how much you originally put in
and to any interest that has already been added
over the previous years
This means compound interest grows faster than simple interest
Simple interest is when the same amount of interest is paid in each year
For compound interest you need
a timeframe (e.g. every year for ten years)
a nominal annual rate of interest, r % (e.g. increasing by 3% each year)
What does compounding monthly mean?
If interest of 6% per annum is divided by 12 and paid in every month ( 0.5% each month),
then it is compounding monthly
In general, if r % per annum is paid compounding monthly
then this means
r
% interest is paid in each month
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Examiner Tips and Tricks
Compounding monthly and compounding annually over the same time frame do not
give the same amount of interest - compounding monthly gives more!
What are compounding periods?
Compounding periods are the time intervals after which interest is paid into your
account
e.g. if r % per annum is compounded monthly then
the compounding period is 'a month'
the number of compounding periods per year is 12
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The letter k is used for the number of compounding periods per year
=1
Compounding half-yearly means k = 2
Compounding quarterly means k = 4
Compounding monthly means k = 12
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Compounding annually means k
In general, if r % p.a. (per annum) is paid across k compounding periods per year
then this means
r
% interest is paid in after each compounding period
k
What is the formula for compound interest?
The formula for calculating compound interest is
r ⎞⎟kn
⎟
FV = PV × 1 +
100k ⎠
⎝
⎛
⎜
⎜
Where
FV is the future value
PV is the present value
n is the number of years
k is the number of compounding periods per year
r % is the nominal annual rate of interest
Examiner Tips and Tricks
The formula for compound interest is given in the formula booklet.
Examiner Tips and Tricks
Whilst your GDC may have a finance package, a lot of exam questions are designed
around the formula, so it is often easier to use that.
Worked Example
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Kim invests MYR 2000 (Malaysian Ringgit) in an account that pays a nominal annual
interest rate of 2.5% compounded monthly.
Calculate the amount that Kim will have in her account after 5 years, to the nearest 10
MYR.
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Depreciation
What is depreciation?
Depreciation is when the value of something falls over time
e.g. the price of a car
If the depreciation is occurring at a constant rate (e.g. 5% per year) then it is compound
depreciation
What is the formula for compound depreciation?
The formula for calculating compound depreciation is
r ⎞⎟n
⎟
FV = PV × 1 −
100 ⎠
⎝
⎛
⎜
⎜
Where
FV is the future value
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PV is the present value
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n is the number of years
r % is the rate of depreciation
Examiner Tips and Tricks
The formula for compound depreciation is not given in the formula booklet, but it can
be found from the compound interest formula by
=1
+ to −
setting k
changing
Examiner Tips and Tricks
If you are using the finance package on your GDC for a depreciation question, make
the interest rate negative.
Worked Example
Kyle buys a new car for AUD $14 999. The value of the car depreciates by 15% each
year.
(a) Find the value of the car after 5 years.
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(b) Find the number of years and months it will take for the value of the car to be
approximately AUD $9999.
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