MCR3U
Unit #7: Sequences and Series Lesson #1: Arithmetic Sequences
Lesson 7.1 Arithmetic Sequences
To begin our discussion of sequences and series, we should define a few terms:
Sequence: A sequence is a list of things usually numbers that are in some specific
order for example 2, 4, 6, 8.
Arithmetic Sequence: An arithmetic sequence is one that has the same difference 𝑑
(called the common difference), between any pair of consecutive terms. The example
above is an arithmetic sequence because the common difference 𝑑 = 2.
Term: Each number in the sequence is called a term. For example, term 1, t1=2 in the
sequence above. Note that a subscript is used to denote the number of the term. The
first term is also denoted with letter 𝑎.
Rule: The formula that can be used to generate any of the terms is called a rule. For
example, for the sequence above, 𝑡𝑛 = 2𝑛 where 𝑛 ∈ 𝑁, 𝑛 > 0.
General Term: The general term is labelled tn. It is usually generated by a formula or
rule. The general term for an arithmetic sequence can be found using the rule
𝒕𝒏 = 𝒂 + (𝒏 − 𝟏)𝒅 where 𝑎 is the first term and 𝑑 is the common difference.
Ex. #1 For the sequence below:
3, 8, 13, 18, 23, 28, 33, 38, …
a) Is this an arithmetic sequence? Explain your answer
b) Find the common difference, d
c) Use the first term, a and the common difference to find the general term, tn
d) Use the general term to find t50.
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MCR3U
Unit #7: Sequences and Series Lesson #1: Arithmetic Sequences
Recursive Sequence: A sequence for which one term is given and the successive
terms can be generated from the previous term.
Recursive Formula: A recursive formula is another way to generate any term if you
know the previous term(s). In general, the first term is 𝑎 so 𝑡1 = 𝑎 and 𝑡𝑛 = 𝑡𝑛−1 + 𝑑. For
the sequence above, the recursive formula would be 𝑡1 = 2, 𝑡𝑛 = 𝑡𝑛−1 + 2 where 𝑛 ∈
𝑁, 𝑛 > 1.
e) Determine the recursive formula for the sequence from page 1
f) Use the recursive formula to find the next three terms
Ex. #2. For the sequence below:
3, 12, 21, 30, …
a) Is this an arithmetic sequence? Explain your answer
b) Find the common difference
c) Determine the recursive formula
d) Use the recursive formula to find the next three terms
e) Find the general term
f) Use the general term to find t50.
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MCR3U
Unit #7: Sequences and Series Lesson #1: Arithmetic Sequences
Eg. #3. A recursive formula for a particular sequence is 𝑡1 = 1, 𝑡𝑛 = 𝑡𝑛−1 + 2 where 𝑛 ∈
𝑁 and 𝑛 > 1.
a) Is this an arithmetic sequence? Explain your answer
b) Find the common difference
c) Use the recursive formula to find the first four terms
d) Find the general term
e) Use the general term to find t50.
Eg. #4. What is the 50th term of the sequence 18, 11, 4, -3, …?
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MCR3U
Unit #7: Sequences and Series Lesson #1: Arithmetic Sequences
A sequence can also be represented as a discrete linear function 𝑓(𝑛) = 𝑑𝑛 + 𝑏 where
𝑏 = 𝑡0 = 𝑎 − 𝑑
Eg. #5. For the sequence 5, 7, 9, …
a) Graph the terms of the arithmetic sequence
b) Use the graph to represent the sequence
as a discrete linear function.
c) Find 𝑡50 .
Eg. #6. Terry invests $300 in a G.I.C. that pays 6% simple interest per year. When will
his investment be worth $732?
Eg. #7. The 7th term of an arithmetic sequence is 53 and the 11th term is 97. Determine
the 100th term.
Homework:
pp. 424 – 425 # 1 – 4, 5ace, 7ac, 8ace, 9ac.
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