Download Sequences and Partial Sums

MATH 1111 – Chapter 12
Sequences and Partial Sums
Definition: A sequence is a function f whose domain is the set of natural numbers. The
values f (1), f (2), f (3), . . . are called the terms of the sequence.
Example 1: Find the first five terms and the 100th term of the sequence defined by each
formula.
(a) an = 2n − 1
(b) cn = n2 − 1
(c) rn =
(−1)n
2n
Definition: An arithmetic sequence is a sequence of the form
a, a + d, a + 2d, a + 3d, . . .
The number a is the first term and d is the common difference of the sequence. The
nth term of an arithmetic sequence is given by
an = a + (n − 1)d
1
Example 2: Find the first six terms and the 300th term of the arithmetic sequence
(a) 2, 5, . . .
(b) 13, 7, . . .
(c) given by a = 1 and d = 4
Definition: A geometric sequence is a sequence of the form
a, ar, ar2 , ar3 , . . .
The number a is the first term, and r is the common ratio of the sequence. The nth
term of a geometric sequence is given by
an = arn−1
Example 3: Find the first five terms and the 250th term of the geometric sequence
(a) 5, 15, 45, . . .
(b) 1, 31 , 91 , . . .
(c) given by a = 2 and r = −5
2
Definition: For the sequence
a1 , a2 , a3 , . . . , an , . . .
the partial sums are
S1 = a1
S2 = a1 + a2
S3 = a1 + a2 + a3
..
.
Sn = a1 + a2 + a3 + · · · + an
..
.
S1 is called the first partial sum, S2 is the second partial sum, and so on. Sn is called
the nth partial sum. The sequence S1 , S2 , S3 , . . . is called the sequence of partial sums.
Example 4: Find S1 , S2 , and S3 for the sequences in Example 1.
Theorem: Partials Sums of an Arithmetic Sequence are given by the following formulas.
n
(2a + (n − 1)d)
2
a + an
2. Sn = n
2
1. Sn =
Partial Sums of a Geometric Sequence are given by the formula
1. Sn = a
1 − rn
1−r
3
Example 5: Find S3 and Sn for the sequences in Examples 2 and 3.
4