SEQUENCES AND SERIES
ARITHMETIC AND GEOMETRIC PROGRESSIONS

Objectives:

• Find the terms of a sequence given the general term.
• Use sequences to solve applied problem.
• Use summation notation to evaluate a series.
• Determine whether a series is arithmetic or geometric.
• Find the sum of a finite or infinite arithmetic/geometric sequence.
SEQUENCES

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.

Example 1.
The sequence:

is an example of a sequence defined by a formula. Its first four terms are
x1=3/8, x2=9/10, x3=67/24, x4=351/70.

Example 2
The sequence

is an example of a sequence defined by recursion (alternatively, we say it is defined inductively). Its first four terms are: x1=0.8, x2=-0.36, x3=-0.87, x4=-0.24

The terms of a sequence need not all be positive, as shown in Example 2.

Other examples of sequences
* 3,5,7,9,11,...
* 1,2,6,24,120, ...
* 1,-1,1,-1,1,...

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.

Solution.  First we enter (into our calculator) the number 1/2 . The output will be 0.5
Then we add to it the next number on the list to get

Again we add to it the next number on the list

We keep on doing this. For example we have

So it seems  the numbers are getting closer and closer to 1. We will see (in geometric series) that this is the correct answer.

Example 4  If f(i) represents some expression (function) involving i, then

has the following meaning:

The "i=" part underneath the summation sign tells you which number to first plug into the given expression. The number on top of the summation sign tells you the last number to plug into the given expression. You always increase by one at each successive step. For example,

=3+6+11+18=38.

Summation rules

Properites of the sums If f(i) represents some expression involving i, then

Example 5  Evaluate

Solution:

ARITHMETIC SEQUENCES AND SERIES

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.

Example 6

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

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

Example 7 Find the formula for the nth term of an arithmetic sequence whose common difference is 3 and whose first term is 2.

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,. . .

The Sum of a Finite Arithmetic Sequence

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
2*S = (a1 + an)n
So,
(a1 + an)n
S = ----------------
2

Example 8 Find the sum:  1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19.

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.

GEOMETRIC SEQUENCES AND SERIES

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.

Example 9.

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

The nth term of a geometric sequence has the form

where r is the common ratio of consecutive terms of the sequence.

Example 11  Write the first five terms of the geometric sequence whose first term is a1=3 and whose common ratio is r=2

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

Example 12 Find the 15th term of the geometric sequence whose first term is 20 and whose common ratio is 1.05

Solution: We use the formula of the general term of a geometric sequence

So a15=20*(1.05)^14
a15=39.599

The Sum of a Geometric Sequence

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.

Example 13 Find the sum of the first 10 terms of the geometric series: 4, 8, 16, 32, 64, . . .

Solution: a1=4. r=2, so a10=4(2^9)=2048. Therefore
4(2^10 - 1)
S10  = ----------------   =  4*1023 = 4092
(2 - 1)

The Sum of an Infinite Geometric Sequence

If |r|<1, the infinite geometric series
a1, a1*r, a2*r^2, . . . , a1*r^n-1, . . .
has the sum
a1
S = ----------------
1 - r

Example 14. The geometric series

1+1/2+1/4+1/8+1/16...

has the sum  S=1/(1-1/2), so S=2.

Example 15  The sum of the infinite series with the general term 4(0.6)^n-1 is
4
S = ----------------=10
1 - 0.6

Note Important differences

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!!

EXERCISES

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 +. . .