Ch. 13
SEQUENCES, ARITHMETIC, AND GEOMETRIC FORMULAS
Sec. 13.1: A sequence is a function whose domain is the set of positive integers.
The function values: a1, a2, a3, . . . an are the terms of the sequence.
a1 = 1st term
n = your number of terms
an = your last term
The notation for the sum of the terms of a finite sequence is called a summation notation or
sigma notation, because it uses the Greek letter sigma.
n
∑ ak = a1 + a2 +. . . an
k=1
k = is called the index of summation, sometimes the letter i is used.
n = is the upper limit of the summation
1 = is the lower limit of the summation (The lower limit does not have to start at 1)
This is read “the sum from k = 1 to n of ak”
Formulas for Sequences:
Examples:
n
∑ c = c + c + c . . . +cn = 𝐂n
𝐂 is a constant
k=1
n
∑ k = 1 + 2 + 3+. . . +n =
k=1
n
∑ k 2 = 12 + 22 + . . . n2 =
k=1
n
n(n + 1)
2
n(n + 1)(2n + 1)
6
n
∑ Ca k = C ∑ a k
k=1
n
k=1
n
𝑛
∑ ak ± bk = ∑ ak ± ∑ bk
𝑘=1
k=1
𝑘=1
Sec. 13.2 : Arithmetic Sequence is a sequence with a common difference.
* * To find a specific term in an arithmetic sequence:
an = a1 + (n – 1) d
an = the nth term in a series
a1 = 1st term
n = the number of the term
d = common difference between each term in an arithmetic series
* * To find the sum of an arthimetic series :
Sn =
n
2
(a1 + an)
a1 = 1st term
n = # of terms
an = last term
Sec. 13.3 : Geometric Sequence is a sequence with a common ratio.
* * To find the nth term in a geometric sequence :
an = a1 rn-1 , r ≠ 0
r = common ratio
* * To find the sum of the first n terms of a geometric sequence :
Sn = a 1 ∗
1−rn
1−r
* * Amount of an Annuity :
P is the deposit made at the end of each payment period for an annuity paying i percent
interest compounded c per payment period. Amount of an annuity A after n deposits.
𝑖
((1 + )n − 1)
c
A=P
i
( )
c