SEQUENCES AND SERIES
ARITHMETIC AND GEOMETRIC
PROGRESSIONS
Objectives:
Definition of Sequence. An infinite sequence is a function whose domain is the set of positive integers. The function values
are the terms of the sequence.. If the domain of the function consists of the first n positive integers only, the sequence is called finite sequence.
The most obvious example of a sequence is the sequence of natural numbers. Note that the integers are not a sequence, although we can turn them into a sequence in many ways; for example by enumerating them as 0, 1, -1, 2, -2.... Here are some more sequences.
The terms of a sequence need not all be positive, as shown in Example 2.
The elements of a sequence are
called the terms.
The 'n-th
term' or 'general term'
of the first example is (2n
+ 1).
The sequence is completely determined by
this general
term. Therefore we write the first sequence as {2n +
1}.
The second sequence is {1*2*3*4*...*n} or
{n!}.
The third is
{(-1)^(n+1) }
SERIES AND SUMMATION NOTATION
Definition of
Summation Notation. Let
be a sequence. We call the sum
an infinite series
(or
just a series) and denote it as
where n is called the index of
summation.We
define a second sequence, 
, called the
partial sums,
by
or, in
general,
.
where i is called
index of summation,n
is the upper limit of
summation and 1 is the lower
limit of
summation.
Properites of the sums If
f(i) represents
some expression involving i, then
Definition of Arithmetic
Sequence. A sequence
is arithmetic if the difference
between
any two consecutive terms is the same. Thus, a sequence is
arithmetic
if
there is a number
d such that 
for any n positive integer greater than 1.The number d is
the common
difference
of the
arithmetic sequence.
a. The sequence whose nth term
is 4n+3 is
arithmetic. For this sequence, the common difference between
consecutive
terms is 4.
7, 11, 15, 19, . . .
4n+3, . . .
b. The sequence whose
nth term is 7-5n is
arithmetic. For this sequence, the common
difference between consecutive
terms is -5.
2,
-3, -5, -18, . . ., 7-5n, . . .
c. The
sequence whose nth term is (n+3)/4 is arithmetic.
For this
sequence, the common difference between consecutive terms is
1/4.
1, 5/4, 3/2, . . .,(n+3)/4, . .
.
The nth term of an
arithmetic sequence has the
form
where d is the common difference between consecutive
terms
of the sequence and c=a1-d.
If you
substitute a1-d for c in the formula
above, then the nth
term of an arithmetic sequence has the alternative
recursion
formula
Solution: We must have an=3n+c.
Since a1=2, it follows that c=a1-d=2-3=-1.
So, the formula for the nth term is
an=3n-1.
The sequence therefore has the
following form:
2,5,8,11,14, . . .,3n-1,. .
.
Say
S = a1 + a2 + ... + an , then S = a1 + a1 +d
+ a1 + 2d + ... +
an
Now write the same sequence in reverse order S
= an +
an - d + an - 2d + ... + a1
Addition
gives
2*S = (a1 + an)n
So,
(a1
+ an)n
S = ----------------
2
Solution: To begin, notice that the series is
arithmetic
(with the common difference of 2). Moreover, the sum has 10
terms. So, the
sum of the series is
S=1 + 3 + 5 + 7 + 9 + 11 +
13 + 15 + 17 + 19
(1 + 19)10
S = ----------------
2
S= 5(20)
S=100.
Consecutive terms of a geometric sequence have a common ratio.
Definition of Geometric Sequence. A sequence
is
geometric if the ratio
of
any two consecutive terms is the same. Thus a sequence is geometric if
there
is
a number r different from 0, such that
for any n positive integer greater than 1. The number r is the
common ratio of the
geometric
sequence.
a. The sequence whose
nth
term is 2^n is
geometric. For this sequence, the common ratio between
consecutive terms
is 2.
2, 4, 8, 16, . . ., 2^n,
. . .
b. The sequence whose nth term
is 4(3^n)
is geometric. For this sequence, the common ratio between
consecutive terms
is 3.
12, 36, 108, 324, . . .,
4(3^n), . . .
c. The sequence whose nth
term is(-1/3)^n is geometric.
For this sequence, the common ratio between
consecutive terms is -1/3.
-1/3, 1/9, -1/27, . .
. (-1/3)^n, . . .
The nth
term of a geometric sequence has the form
where r is the common ratio of consecutive terms of
the
sequence.
Solution: Starting with 3 repeatedly multiply by
2 to
obtain the following:
a1=3
a2=3(2^1)=6
a3=3(2^2)=12
a4=3(2^3)=24
a5=3(2^4)=28
Solution: We use the formula of the
general term
of a geometric sequence
So a15=20*(1.05)^14
a15=39.599
Here is the technique for calculating the sum of the first n terms of a geometric sequence.
S = a1 + a2 + ... + an , then
=> S*r = a1*r + a2*r + ... + an*r
=> S*r = a2 + ... + an + an*r
=> S*r - S = an*r - a1
=> S(r-1) = an*r -a1
an*r
- a1 a1*(r^n)
-
a1
=> S =
----------------
= ----------------
(r
-
1)
(r
- 1)
a1*(r^n
- 1)
=> S =
----------------
(r
- 1)
where r is the common ratio different from
0.
Solution: a1=4. r=2, so a10=4(2^9)=2048.
Therefore
4(2^10
- 1)
S10 =
----------------
= 4*1023 = 4092
(2
- 1)
If |r|<1, the
infinite geometric series
a1, a1*r, a2*r^2, . . .
, a1*r^n-1, . . .
has the sum
a1
S = ----------------
1
- r
1+1/2+1/4+1/8+1/16...
has the sum S=1/(1-1/2), so S=2.
Finite sequence: 1, 5, 9, 13, 17
Finite
series: 1 + 5 + 9 + 13
Infinite sequence: 1, 2,
4, 8, 16, . . .
Infinite series: 1 + 2 + 4 + 8 +
16 + . . .
Note that a series is an indicated sum of the terms of a sequence!!
1. Find f(2) for f(n)=2n/(n^2+1)
2. Find
the sums
3. Evaluate
4. Find the formula for the general term of the
arithmetic
sequence with:
a) a1=4, d=5;
b) a1=-2, d=3
c) 2, 8, 14, 20, 26, . .
.
d) a1=2, a12=46
5. Find the formula for an only if it is an arithmetic
sequence:
3, 7, 11, 15, 19, . . .
6. Give the
formula for an only if it is a geometric sequence:
2, 5, 10, 17, 26, 37,
. . .
7. Find the recursive formula
for the sequence: 3, 13,
33, 73, 153, . . .
8. Find the sum of the first 50 terms of the arithmetic
series:
6+12+18+24+30, . . .
9. Find the sum
of all the multiples of 4 between 1 and
999.
10. Find the sum of the series 3/4 + 3/8 + 3/16 + 3/32
+. . .
.
11. Find the sum of the first 100
terms of the sequence
with the general term 2n+5
.
12. Find the sum of the first 100 odd
integers.
13. Find the 5th term of the
geometric sequence with a1=16
and r=1/4.
14. Find the sum of the first 9 terms of the sequence
with
the general term (-2)^n.
15. Find
the sum, if it exists:
a) 4 + 1 + 1/4 + 1/16 +. .
.
b) 5 + 20/3 + 880/9 + 320/27 +. .
.