Lecture Notes - Sequences

Sequences Lecture Notes for Section 8.1

A sequence is an infinite list of numbers written in a definite order: #ß

%ß

)ß

"'ß

$#ß

á

The numbers in the list are called the terms of the sequence. In the sequence above, the first term is #, the second term is %, the third term is ), and so forth, with each successive term being twice the previous term. How can we figure out the 10th term in this sequence? Well, we could simply continue doubling until we arrive at the tenth term: #ß

%ß

)ß

"'ß

$#ß

'%ß

"#)ß

#&'ß

&"#ß

"!#%

However, a better method would be to find a formula for the sequence, i.e. a formula for how the 8th term depends on 8. In this case, each term is a power of #: #" œ #ß

## œ %ß

#$ œ )ß

#% œ "'ß

#& œ $#ß

á

In particular, the formula for the 8th term of this sequence is #8 . Thus the 10th term must be #"! , which is "!#%. Different Ways of Writing a Sequence It's often clearer when writing a sequence to provide a formula for the 8th term immediately. One method is to include the formula among the list of terms: #ß

%ß

)ß

"'ß

áß

#8 ß

á

Sometimes, it is convenient to write only the formula for a sequence. The convention is that any formula surrounded by braces specifies a sequence: e#8 f_ 8œ"

or simply

e#8 f.

There is also a convention for discussing sequences abstractly. sequence in general, we will write the terms using variables: +" ß

+# ß

+$ ß

+% ß

When talking about a

á

To avoid running out of different letters, we use the same letter for all the variables (in this case +), with subscripts to distinguish between different terms. Such a sequence may also be written using braces: e+ 8 f_ 8œ"

or simply

e+ 8 f.

Formulas for Sequences The trick to finding the formula for a sequence is to recognize the pattern, and figure out how to describe it in terms of 8. Here are a few simple examples: EXAMPLE 1 Find formulas for the following sequences: (a)

"ß

" " " " ß ß ß ß á # $ % &

%ß &ß 'ß (ß )ß á

(b)

(c)

"ß %ß *ß "'ß #&ß á

SOLUTION

(a) This is the sequence e"Î8f. (b) Usually a good way of figuring out the formula is to make a table showing 8 and +8 : 8 +8

" %

# &

$ '

% (

& )

â â

As you can see, +8 is always three greater than 8, so this is the sequence e8  $f. (c) We make a table showing 8 and +8 : 8 +8

" "

# %

$ *

% "'

& #&

â â

As you can see, the 8th term is equal to 8# , so this is the sequence e8# f.

è

There are certain sequences that you should know on sight:

COMMON SEQUENCES e#8 f: e$8 f:

e8 # f: e8 $ f: e8xf:

#ß

%ß

)ß

$ß

*ß

#(ß

"ß

%ß

*ß

"ß

)ß

#(ß

"ß

#ß

'ß

"'ß )"ß "'ß '%ß #%ß

$#ß #%$ß #&ß "#&ß "#!ß

'%ß

á

(#*ß $'ß

á

#"'ß (#!ß

á

á á

The last of these is the sequence of factorials, which you may not be familiar with. The 8th term in this sequence (written 8x, and pronounced “8 factorial”) is the product of all the whole numbers between " and 8. For example: &x œ " ‚ # ‚ $ ‚ % ‚ & œ "#!.

EXAMPLE 2 Find formulas for the following sequences: * #( )" #%$ ß ß ß ß á # ' #% "#!

(a)

$ß

(b)

È$ß %ß $È&ß %È'ß &È(ß 'È)ß á

(c)

"'ß #&ß $'ß %*ß '%ß á

SOLUTION

(a) This is the sequence œ

$8 . 8x

(c) Let's compare +8 with 8: 8 +8

" È$

# %

$ È $ &

% È % '

& È & (

' È ' )

â â

For the latter three terms, the coefficient is 8, and the number inside the square root is 8  #. This formula also works for the first and second terms: È $ œ "È $

Therefore, this is the sequence ˜8È8  #™.

% œ #È %

(d) Each of the terms in this sequence is a perfect square. Indeed: 8 +8

" "' œ %#

# #& œ &#

$ $' œ '#

% %* œ (#

& '% œ )#

â â

The number being squared is always 8  $, so this is the sequence ˜a8  $b# ™.

è

Special Sequences Two types of sequences that we will encounter repeatedly are arithmetic sequences and geometric sequences. An arithmetic sequence is a sequence for which each term is a constant plus the previous term. For example, in the sequence &ß

)ß

""ß

"%ß

"(ß

á

each term is obtained from the previous term by adding $. This number $ is called the common difference, since it can be obtained from subtracting any two consecutive terms. The formula for an arithmetic sequence is always a linear function:

ARITHMETIC SEQUENCES If e+8 f is an arithmetic sequence with common difference . , then +8 œ 5  8. for some constant 5 . EXAMPLE 3 Find a formula for the sequence e&ß )ß ""ß "%ß "(ß á f. SOLUTION

Since the common difference is $, the formula for this arithmetic sequence must

have the form +8 œ 5  $8 where 5 is some constant. Since +" is supposed to be &, the constant 5 must be #. Therefore, this is the sequence e#  $8f. è A geometric sequence is a sequence for which each term is a constant multiplied by the previous term. For example, in the sequence 'ß

"#ß

#%ß

%)ß

*'ß

á

each term is exactly # times the previous term. The number # is called the common ratio, since it can be obtained by taking the ratio of any two consecutive terms. The formula for a geometric sequence is always an exponential function:

GEOMETRIC SEQUENCES If e+8 f is a geometric sequence with common ratio