TOPIC 1 – ALGEBRA
ARITHMETIC SEQUENCES & SERIES
S. Aldous, A. Beetz & S. Thauvette
IB DP SL Mathematics
You Should Be Able To…
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State whether a sequence is arithmetic, giving an
appropriate reason
Find the common difference in an arithmetic
sequence
Find the nth term of an arithmetic sequence
Find the number of terms in an arithmetic sequence
Solve real-world problems involving arithmetic
sequences and series.
Challenge – Nob’s Tricky Sequence
Nob Yoshigahara discovered this beautiful number
sequence. Can you work out the logic behind the
sequence and fill in the missing number?
Make some sequences by picking four numbers that
form a pattern. Record as many as you can.
How do you see this pattern growing?
Draw shapes 4 and 5.
How many matchsticks are in shape 10?
Can you describe the pattern using algebra?
Finding the General Term
Use 2 pieces of paper.
On one, make up a value for u1.
On the other, make up a value for u4.
Swap the cards with someone else.
Find the general term for the arithmetic sequence.
Make sure you both agree.
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Each day a runner trains for a 10km race. On the
first day she runs 1,000m, and then increases the
distance by 250m each subsequent day.
 On
which day does she run a distance of 10km in
training?
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In an arithmetic sequence, the first term is –2, the
fourth term is 16, and the nth term is 11,998.
 Find
the common difference d.
 Find the value of n
Question – Finding Un Given Two Terms
In an arithmetic sequence, U7 = 121 and U15 = 193.
Find the first three terms of the sequence and Un.
Substitute know values in the formula for the nth term to
write a system of equations. Then, solve the system.
Since a = 67 and d = 9, the first three terms of the
sequence are 67, 76, and 85.
Finding Un Given Two Terms continued…
To find Un , substitute 67 for a and 9 for d in the formula
for the nth term.
Un = 67 + (n – 1)9
Un = 67 + 9n – 9
Un = 9n + 58
Thus, the first three terms are 67, 76, and 85, and
Un = 9n + 58.
You Should Know…
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A sequence is arithmetic if the difference between
consecutive terms is the same
An arithmetic sequence has the form:
u1, u1 + d, u1 + 2d, u1 + 3d, …, u1 + (n – 1)d
The common difference can be found by
subtracting a term from the subsequent term:
d = un + 1 – un
When to use the term formula
You should know:
Textbook: Arithmetic Sequences p.155 – 159
Homework: Arithmetic Sequences
ARITHMETIC SERIES
S. Aldous, A. Beetz & S. Thauvette
IB DP SL Mathematics
Arithmetic Series
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Calculate the sum of the first n terms of an
arithmetic series
Challenge
The top three layers of boxes in a store display are
arranged as shown. If the pattern continues, and there
are 12 layers in the display, what is the total number
of boxes in the display?
Treasure Hunt
In the pod there are ten pink cards.
Find any card. Note down its number.
Solve the question on the card.
Find the answer on another card somewhere in the pod.
Note down the card’s number.
Continue answering questions and noting the card
numbers. You should finish at the same card you started.
Show your teacher the list of card numbers you visited.
Sum of a Series Given First Terms
Find the sum of the first 60 terms of the series:
(a) 5 + 8 + 11 + …
Sum of a Series Given First and Last Terms
Consider the series 17, 7, –3, …, –303.
(a) Show that the series is arithmetic.
Show that the difference between two consecutive terms
is constant. For example:
7 – 17 = –3 – 7 = –10
Therefore, d = –10 and the series is arithmetic
Continued…
Consider the series 17, 7, –3, …, –303.
(b) Find the sum of the series.
The formula for the sum of an arithmetic series requires
the value of n. Use the term formula first to find n.
n = 33
Now use the appropriate formula to find the sum of the
first 33 terms.
S33 = –4719
Question
The sum of the first five terms of an arithmetic series is
65/2. Also, five times the 7th term is the same as six
times the second term. Find the first term and common
difference.
Question continued…
Be Prepared
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Look for words or expressions that suggest the use
of the term formula—”after the 10th month”, “in the
8th row”—and those that suggest the sum formula—
”total cost”, “total distance”, “altogether”.
Look for questions in which information is given
about two terms. This normally suggests the
formation of a pair of simultaneous equations that
you will have to solve to find the first term and the
common difference.
The last term of a sequence can be used to find the
number of terms in the sequence
You should know:
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When to use the sum formula
Textbook: Arithmetic Series p.167 – 169
Homework: Arithmetic Series