Download Sequences and Series

SEQUENCES AND SERIES
Sequence: a function whose domain is the set of natural numbers (the term numbers) and
whose range is the set of term values (patterns).
Example #1: Arithmetic Sequence
1
2
3
4
5
6
3
10
17
24
31
38
An arithmetic sequence (progression: each term after the first is obtained by adding or
subtracting a fixed number, called the common difference, to the preceding term.
Example #2: Geometric Sequence
1
2
3
4
5
6
3
6
12
24
48
96
A geometric sequence: each term after the first is the product of the preceding term and
a fixed number (called the common ratio).
Example #3: Fibonacci Sequence
1
2
3
4
5
6
1
1
2
3
5
8
A Fibonacci sequence: beginning with the third term, the value in each term is equal to
the sum of the two preceding values. Example: f(6) = 8 = f(4) + f(5) = 3 + 5 = 8.
SERIES
A series is the indicated sum of the terms in a sequence.
Example #10: Calculate the sum of numbers in the following series:
1 + 2 + 3 + 4 + … + 97 + 98 + 99 + 100
If we add the sums of the outside pairs, the second and ninety-ninth pairs, and so on, we
have 50 such pairings that equal 101. So the sum would be equal to (50)(101) = 5050.
Arithmetic Series: The sum of a finite arithmetic sequence
a1 + a2 + a3 + ... + an
S n = Sum of the n terms in the sequence
Sn =
a is the first term in the sequence
n
(a1 + an ) . 1
2
a n is the last term in the sequence
n = the number of terms in the sequence
Example #11: Find the sum of the series in example #10.
a1 =1
a n = 100
Sn =
n = 100
100
(1 + 100) = (50)(101) = 5050
2
Alternate formula: S =
n
[2a1 + (n ! 1)d ] (you do not know the last term.)
2
Example #12: Find the sum of the first ten terms of the arithmetic series
3 + 8 + 13 +…
a1 =3
d =5
n = 10
S=
10
[2(3) + (10 ! 1)5] = (5)(6 + 45) = 255
2
is
We use the Greek letter
! (sigma) to indicate a sum.
5
S=
" (2n ! 1) = 2(1)-1 + 2(2)-1 + 2(3)-1 + 2(4) -1 + 2(5) -1= 25
n =1
Where n is the lower limit or least value of n, 5 is the upper limit or greatest value of n,
and (2n – 1) is the general term or explicit formula.
Example #13:
Evaluate
10
! 2k + 7 = 9 + 11 + 13 + …+ 27
k =1
S10 =
10
(9 + 27) = (5)(36) = 180
2
Geometric Series: The sum of the terms of a finite geometric sequence is
a1 (1 ! r n )
Sn =
1! r
where
a1
is the first term, r is the common ratio, and n is the number
of terms.
Example #14: Evaluate the series to the given term.
S6
for 2 + 6 + 18 + …
S6 =
2(1 ! 36 ) 2(1 ! 729)
=
= (!1)(!728) = 728 1! 3
!2