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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 49 "High School Modul
es > Algebra by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 "    " 
}{TEXT 256 9 "Sequences" }}{PARA 0 "" 0 "" {TEXT -1 132 "\nExploration
 of simple sequences - arithmetic, geometrc, and other  - various prop
erties and computations, and graphs of sequences.\n" }}{PARA 0 "" 0 "
" {TEXT 258 153 "[Directions : Execute the Code Resource section first
. Although there will be no output immediately, these definitions are \+
used later in this worksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 " 0. Code" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 9 "restart; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 121 "GeoSum :=proc(a,N)\n    local sum, k;\n    sum := 0:\n    for k
 from 0 to N do sum := sum + a(k); od:\n    sum;\n    end proc:" }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 29 " 1.  The Concept of Sequences" }}
{PARA 0 "" 0 "" {TEXT -1 94 "\nThe idea of a sequence is a list of num
bers, for example the squares of the natural numbers.\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "k^2 $ k = 1..10;" }}}{PARA 0 "" 0 "
" {TEXT -1 100 "\nWe can think of these numbers as being the range of \+
a function, whos domain is the natural numbers." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 49 "f := k -> k^2;\narray( [[ k,f(k) ] $ k = 1..10
] );" }}}{PARA 0 "" 0 "" {TEXT -1 62 "\nThe 7th member of this list is
 f(7), and the 777th is f(777);" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "f(7); f(777);" }}}{PARA 0 "" 0 "" {TEXT -1 177 "\nUsually we d
on't use standard function notation because that makes us think about \+
domains which include non-natural numbers. Often we call sequences by \+
letters with subscripts." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "
for k from 1 to 10 do a[k] = k^2; od;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "a[7] = f(7); a[77] = f(77);" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 25 " 2.  Arithmetic \+
Sequences" }}{PARA 0 "" 0 "" {TEXT -1 7 "\n      " }{TEXT 262 36 "Defi
nition of an Arithmetic Sequence" }{TEXT -1 168 "\n\nAn arithmetic seq
uence is a list of numbers which have a common difference d. The entir
e sequence is determined by the starting value a[1] and the common dif
ference.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "k := 'k': a :=
 'a':\na[k] := a[1] + d*(k-1);" }}}{PARA 0 "" 0 "" {TEXT -1 23 "\nHere
 are some examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "b :=  \+
k -> 9 + 7*(k-1);   \nb(k) $ k = 1..20;" }}}{PARA 0 "" 0 "" {TEXT -1 
102 "\nIf we take the difference of any two consecutive members of the
 sequence we get the common differnce." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 54 "b(4) - b(3);      b(11) - b(10);      b(100) - b(99);
 " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "c :=  k -> -11 + 5*(k-
1);\nc(k) $ k = 1..20;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "c
(4) - c(3);      c(11) - c(10);      c(100) - c(99); " }}}{PARA 0 "" 
0 "" {TEXT -1 54 "\n\nYou may recognize these common arithemetic seque
nces" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "b :=  k -> 1 + 2*(k-
1);   \nb(k) $ k = 1..20;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
44 "b :=  k -> 0 + 2*(k-1);   \nb(k) $ k = 1..20;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 45 "b :=  k -> 0 + 3*(k-1);   \nb(k) $ k = 11..2
0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "b :=  k -> 5 + 5*(k-1
);   \nb(k) $ k = 1..20;" }}}{PARA 0 "" 0 "" {TEXT -1 7 "\n      " }
{TEXT 261 20 "Finding the nth Term" }{TEXT -1 136 "\n\nIts easy to fin
d any term of the sequence if you know the formula. Since the formula \+
is a function, we simply evaluate the function. \n" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 24 "a := k -> 100 + (k-1)*9;" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 86 "`the 9th member = `, a(9);\n`the 99th membe
r = `, a(99);\n`the 999th member = `, a(999);" }}}{PARA 0 "" 0 "" 
{TEXT -1 191 " \n\nA common problem is to find a certain term in the s
equence WITHOUT being given the formula. However, we can easily find t
he formula if we know at least two members of the sequence.\n\n      \+
" }{TEXT 263 19 "Finding the Formula" }{TEXT -1 2 "\n\n" }{TEXT 264 7 
"Problem" }{TEXT -1 4 " :  " }{TEXT 265 71 "If an arithmetic sequence \+
begins  2, 9, 16, ... , find the 1000th term." }{TEXT -1 64 "\n\nClear
ly a[0] = 2. We only need to find the common difference.\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "d = 9-2;" }}}{PARA 0 "" 0 "" {TEXT 
-1 20 "\nHere is the formula" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
22 "a := k -> 2 + 7*(k-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "a(1000);" }}}{PARA 0 "" 0 "" {TEXT -1 8 " \n      " }{TEXT 259 15 "
Finding the Sum" }{TEXT -1 43 "\n\nHere is how to find the sum of a se
ries.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a := k -> 16 + 9*
(k-1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "sum( a(k), k = 1.
.100);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 
"" {TEXT -1 7 "\n      " }{TEXT 260 25 "Finding the Average Value" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "a := k -> 17 + 10*(k-1);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sum( a(k), k = 1..100);\n%/
100;" }}}{PARA 0 "" 0 "" {TEXT -1 130 "\nSo we computed the sum and di
vided by the number of terms. However, look at these interesting facts
 which also give the average.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "( a(50) + a(51) ) /2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "( a(1) + a(100) ) / 2;" }}}{PARA 0 "" 0 "" {TEXT -1 153 "\n\nIt \+
turns out that this always works. If you take the average of the first
 and last, its the average of the sequence between and including those
 values." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "N := 1000;\na :=
 k ->  82 + 13*(k-1);\nsum( a(k), k = 1..N);\n%/N;\n( a(1) + a(N) ) / \+
2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "N := 37145;\na := k -
> 52 + 29*(k-1);\nsum( a(k), k = 1..N);\n%/N;\n( a(1) + a(N) ) / 2;" }
}}{PARA 0 "" 0 "" {TEXT -1 40 "\nIt appears to work. Can you prove it?
\n\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 24 " 3.  Geometric Sequences
" }}{PARA 0 "" 0 "" {TEXT -1 7 "\n      " }{TEXT 268 34 "Definition of
 a Geometric Sequence" }{TEXT -1 74 "\n\nA geometric sequence is a lis
t of numbers which have a common quotient, " }{TEXT 274 1 "r" }{TEXT 
-1 58 ". The entire sequence is determined by the starting value " }
{TEXT 272 1 "a" }{TEXT -1 26 " and the common quotient, " }{TEXT 273 
1 "r" }{TEXT -1 2 ". " }{TEXT 275 90 "(Often geometric sequences begin
 their indexing at 0 rather than 1, although this varies.)" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "k := 'k': a := 'a':\na[k] := a*r^k;
" }}}{PARA 0 "" 0 "" {TEXT -1 23 "\nHere are some examples" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "b :=  k -> 100*(1/5)^k;   \nb(k) $ \+
k = 0..10;" }}}{PARA 0 "" 0 "" {TEXT -1 102 "\nIf we take the quotient
 of any two consecutive members of the sequence we get the common quot
ient, r." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "b(4)/b(3);      \+
b(11)/b(10);  b(30)/b(29); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
43 "b :=  k -> 64*(1/4)^k;   \nb(k) $ k = 0..10;" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 44 "b(4)/b(3);      b(11)/b(10);  b(30)/b(29);  \+
" }}}{PARA 0 "" 0 "" {TEXT -1 42 "\n\nHere are some other geometric se
quences." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "b :=  k -> 1*(1/
2)^k;   \nb(k) $ k = 0..10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
44 "b :=  k -> 512*(1/2)^k;   \nb(k) $ k = 0..15;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 43 "b :=  k -> 12*(1/3)^k;   \nb(k) $ k = 0..10;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "b :=  k -> 100*(.01)^k;
   \nb(k) $ k = 0..7;" }}}{PARA 0 "" 0 "" {TEXT -1 120 "\n\nIf 0 < r <
 1, then the geometric sequence decreases toward zero fairly quickly. \+
If r > 1, then the sequence increases." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "b :=  k -> 1*(2)^k;   \nb(k) $ k = 0..14;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "b :=  k -> 100*(5/4)^k;   \nb(k) $ \+
k = 0..7;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "b :=  k -> .00
319*(10)^k;   \nb(k) $ k = 0..7;" }}}{PARA 0 "" 0 "" {TEXT -1 7 "\n   \+
   " }{TEXT 267 20 "Finding the nth Term" }{TEXT -1 136 "\n\nIts easy \+
to find any term of the sequence if you know the formula. Since the fo
rmula is a function, we simply evaluate the function. \n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a := k -> 999*(8/9)^k;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "`the 9th member = `, a(5);\n`the 99
th member = `, a(10);\n`the 999th member = `, a(20);" }}}{PARA 0 "" 0 
"" {TEXT -1 191 " \n\nA common problem is to find a certain term in th
e sequence WITHOUT being given the formula. However, we can easily fin
d the formula if we know at least two members of the sequence.\n\n    \+
  " }{TEXT 269 19 "Finding the Formula" }{TEXT -1 2 "\n\n" }{TEXT 270 
7 "Problem" }{TEXT -1 4 " :  " }{TEXT 271 94 "If a geometric sequence \+
begins  20, 4, ... , find the 12th term (where a[0]=20, a[1] = 4,...).
" }{TEXT -1 60 "\n\nClearly a = 20. We only need to find the common qu
otient.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "d = 4/20;" }}}
{PARA 0 "" 0 "" {TEXT -1 20 "\nHere is the formula" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 21 "a := k -> 20*(1/5)^k;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 6 "a(12);" }}}{PARA 0 "" 0 "" {TEXT -1 8 " \n     \+
 " }{TEXT 266 41 "Formula for the Sum of a Geometric Series" }{TEXT 
-1 117 "\n\nThere is a formula for the sum of a geometric series. We w
ill motivate this formula in a way similar to the proof.\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a:='a':  b:='b':  c:='c':\na:= k ->
 c*r^k;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "S := GeoSum( a, \+
10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "r*S; expand(%);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "S - r*S:  % = expand(%);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "S:= 'S';\nS - r*S = c - c*
r^11 ;\nS*(1 - r) = c*(1-r^11);\nS := c*(1-r^11)/(1 - r);" }}}{PARA 0 
"" 0 "" {TEXT -1 5 "\n    " }{TEXT 276 15 "Finding the Sum" }{TEXT -1 
60 "\n\nThis leads to this formula for a finite geometric series :" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "S := (c, r, N) -> c*(1-r^(N+
1))/(1-r);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "a := k -> 20
*(1/5)^k;\n`first 11 elements = `, a(j) $ j = 0..10;\n`Sum of first 11
 elements = `,S(20, 1/5, 10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 107 "a := k -> 4*(1/2)^k;\n`first 11 elements = `, a(j) $ j = 0..10;
\n`Sum of first 11 elements = `,S(4, 1/2, 10);" }}}{PARA 0 "" 0 "" 
{TEXT -1 6 "\n\n    " }{TEXT 277 15 "Interesting Sum" }{TEXT -1 146 "s
\n\nHere is an interesting case. If we take the simple geometric seque
nce with a = 1 and r = 1/2, and add them up, we get a number very clos
e to 2!" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "a := k -> (1/2)^
k;\n`first 21 elements = `, a(j) $ j = 0..20;\n`Sum of first 21 elemen
ts = `,S(1, 1/2, 20);\nevalf(S(1, 1/2, 20));" }}}{PARA 0 "" 0 "" 
{TEXT -1 337 "\nYou may notice some other coincidences...\n           \+
      .........     r = 1/2  ............    Sum is close to 2\n      \+
           .........     r = 2/3  ............    Sum is close to 3\n \+
                .........     r = 3/4  ............    Sum is close to
 4\n                 .........     r = 9/10  ............  Sum is clos
e to 9" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "a := k -> (2/3)^k
;\n`first 21 elements = `, a(j) $ j = 0..20;\n`Sum of first 21 element
s = `,S(1, 2/3, 20);\nevalf(S(1, 2/3, 20));" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 126 "a := k -> (3/4)^k;\n`first 21 elements = `, a(j) \+
$ j = 0..6;\n`Sum of first 21 elements = `,S(1, 3/4, 20);\nevalf(S(1, \+
3/4, 20));" }}}{PARA 0 "" 0 "" {TEXT -1 30 "\n\nDoes this break the pa
ttern?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "a := k -> (9/10)^
k;\n`first few elements = `, a(j) $ j = 0..6;\n`Sum of first 11 elemen
ts = `,evalf(S(1, 9/10, 10));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 52 "`Sum of first 21 elements = `,evalf(S(1, 9/10, 20));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "`Sum of first 31 elements = `,evalf
(S(1, 9/10, 30));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "`Sum o
f first 41 elements = `,evalf(S(1, 9/10, 40));" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 " 4.  Other Seque
nces" }}{PARA 0 "" 0 "" {TEXT -1 258 "\nThere are an infinite number o
f other varieties of sequences. Here are a few. Note that some of them
 are constructed by combining arithmetic, geometric, factorials, alter
nating signs, and other mathematical operations to create ever more co
mplex sequences.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "k := 'k
':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "a := k -> (-1)^k; a(k
) $ k = 1..20;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "a := k ->
 (-1)^(k+1); a(k) $ k = 1..20;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 42 "a := k -> ((-1)^k)*2^k ; a(k) $ k = 1..15;" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 42 "a := k -> ((-1)^k)/3^k ; a(k) $ k = 1..10;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "a := k -> 1/k! ; a(k) $
 k = 1..10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "a := k -> (k
^2)/k!; a(k) $ k = 1..10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
39 "a := k -> (k^2/2^k) ; a(k) $ k = 1..10;" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 47 "a := k -> (((-1)^k)*k^3)/k! ; a(k) $ k = 1..10;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "a := k -> ((-1)^k)/3^k ; a
(k) $ k = 1..10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "a := k \+
-> 1/(k^2 + 1); a(k) $ k = 1..10;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "a := k -> 1/(k^3 + k!) ; a(k) $ k = 1..10;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "a := k -> lcm(k,k+3) ; a(k) \+
$ k = 1..10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "a := k -> f
loor( k/2) ; a(k) $ k = 1..20;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 51 "a := k -> floor( floor( k/2)/3) ; a(k) $ k = 1..20;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "a := k -> k*2^floor( k/5) ; \+
a(k) $ k = 1..20;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "a := k
 -> gcd(k, 4);   'gcd(k, 4)' $ k = 1..35;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 51 "a := k -> gcd(k, k+5);   'gcd(k, k+5)' $ k = 1..35;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "a := k -> gcd(k, (k-1)!
);   'gcd(k, (k-1)! )' $ k = 1..35;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "a := k -> gcd(k, (k-1)!);   'lcm( gcd(k,4), gcd(k,3) \+
)' $ k = 1..40;" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 
4 "" 0 "" {TEXT -1 36 " 5.  Graphs of Arithemetic Sequences" }}{PARA 
0 "" 0 "" {TEXT -1 289 "\nArithematic sequences have some similarities
 with lines. In both cases, the amount of increase is constant each in
crement in the independent variable. And like lines, arithmetic sequen
ces are increasing or decreasing depending on the coefficient of the v
ariable, which acts like a slope.\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 83 "a := k -> 2*k - 3;\na(k) $ k = 1..10;\nplot( a(floor(
x)) , x = 1..10, thickness = 2);" }}}{PARA 0 "" 0 "" {TEXT -1 75 "\nIn
 fact, if we can plot the sequence and the corresponding line together
 :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot( [a(floor(x)),a(x
)] , x = 1..10, thickness = 2, linestyle = [1,2]);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 88 "a := k -> 5*k + 10;\na(k) $ k = 1..10;\nplo
t( \{a(floor(x)),0\} , x = 1..10, thickness = 2);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 87 "a := k -> 7 - 3*k;\na(k) $ k = 1..10;\nplot(
 \{a(floor(x)),0\} , x = 1..10, thickness = 2);" }}}{PARA 0 "" 0 "" 
{TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 34 " 6.  Graphs o
f Geometric Sequences" }}{PARA 0 "" 0 "" {TEXT -1 71 "\nGeometric sequ
ences have some similarities with exponential functions." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "a := k -> 10*(1/2)^(k);\na(k) $ k =
 1..10;\nplot( a(floor(x)) , x = 0..10, thickness = 2);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot([a(floor(x)),a(x)] , x = 0..10
, thickness = 2, linestyle =[1,2]);" }}}{PARA 0 "" 0 "" {TEXT -1 98 "
\nWhen |r| <  1 then the graph is decreasing as we see above. When r >
 1, the graph is increasing.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 109 "a := k -> 2*(3/2)^k;\na(k) $ k = 1..10;\nplot([a(floor(x)),a(x)
] , x = 0..10, thickness = 2, linestyle =[1,2]);" }}}{PARA 0 "" 0 "" 
{TEXT -1 85 "\nGraphs can help us compare two different sequences. For
 example, look at these two :" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 47 "a := k -> 100*(1/3)^k;    b := k -> 50*(1/2)^k;" }}}{PARA 0 "" 
0 "" {TEXT -1 140 "\nLooking at the graph, we can see that while the a
(k) (the blue graph) starts larger, it decreases faster and becomes sm
aller than b (green)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "plo
t( [a(floor(x)),b(floor(x)),a(x),b(x)] , x = 0..7, thickness = 2, line
style = [1,1,2,2], color = [blue, green,yellow, orange] );" }}}{PARA 
0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 30 " 7.
  Graphs of Other Sequences" }}{PARA 0 "" 0 "" {TEXT -1 37 "\nWe can l
ook at other sequences too.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 82 "a := k -> (-1)^k; a(k) $ k = 1..20;\nplot( a(floor(x)) , x = 1..
20, thickness = 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "a :=
 k -> 1/k; a(k) $ k = 1..20;\nplot( \{a(floor(x)),0\} , x = 1..10, thi
ckness = 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "a := k -> -
((-1)^k)/k; a(k) $ k = 1..20;\nplot( \{a(floor(x)),0\} , x = 1..10, th
ickness = 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "a := k -> \+
k^((-1)^(k+1))  ; a(k) $ k = 1..20;\nplot( \{a(floor(x)),0\} , x = 1..
10, thickness = 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "a :
= k -> 1/ ( k +  3 + ((-1)^(k+1))*2       ); \na(k) $ k = 1..15;\nplot
( \{a(floor(x)),0\} , x = 1..15, thickness = 2);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{PARA 0 "" 0 "" {TEXT 278 36 
"\n         \251 2002 Waterloo Maple Inc " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{MARK "0 1" 36 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }