Math 4
Arithmetic Sequences & Series
Name:__________________________
Date:___________________________
Standards: MA3A9. Students will use sequences and series
a. Use and find recursive and explicit formulae for the terms of sequences.
b. Recognize and use simple arithmetic and geometric sequences.
e. Find and apply the sums of finite and, where appropriate, infinite arithmetic and geometric
series.
f. Use summation notation to explore series.
Sequences: a collection of terms ordered so that it has a first member, second members, etc…
a1, a2, a3, a4, …, an
a) finite sequence – has a set number of terms
b) infinite sequence – # of terms =

You try: Write the first five terms
1. an  5  n
2. f (n)  5n
Arithmetic sequence: A sequence of terms a1 , a2 , a3 ,... where difference between the terms is
constant.
Common difference: In an arithmetic sequence the difference between two consecutive terms is d,
d  an  an1 .
Recursive Sequence: The first term(s) will be given and all other terms will be defined using
previous terms.
Arithmetic Recursive Formula: a1 = value of the first term and an  an 1  d . (both must be given)
Write the first 4 terms of the sequence.
1. a1  3
an  an1  4
2. a1  15
an  2an1  10
Explicit formula: A formula for a sequence that gives a direct method for determining the nth term of
the sequence. It presents the relationship between two quantities, i.e. the term number and the value
of the term.
Arithmetic Explicit Sequence Formula: an  a1  d  n 1
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Write the explicit formula and find the first 4 terms of the sequence.
1. a1  7 , d  3
2. a1  20 , d  4
Find the 15th term of the sequence -17. -13, -9, …
Arithmetic series: The sum of a set of terms in arithmetic sequence a1  a2  a3  ... with d  an  an1 .
a) finite: 3 + 6 + 9 + 12 + 15
b) infinite: 3 + 6 + 9 + 12 + 15 + …
n
Summation or sigma notation:
 a , where i is the index of summation, n is the upper limit of
i 1
i
summation, and 1 is the lower limit of summation. This expression gives the partial sum which is the
sum of the first n terms of a sequence. To find n (the number of terms you are finding the sum of)
use the following : upper limit – lower limit plus 1 (n = UL – LL +1). More generally, we can write
n
 a , where k is the starting value.
i
ik
Sum of a finite arithmetic series: The sum, Sn, of the first n terms of an arithmetic sequence is
n  a1  an 
given by Sn 
, where a1 = value of the first term and an = value of the last term in the
2
sequence.
Properties of Sums
n
1.
 c  cn , c is a constant.
n
2.
i 1
23
1:
1
2:
k 5
i
i 1
10
 2k  11
i 1
n(aLL  aUL )
2
125
i1
3.
i 
13
4.
 13k
k 7
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