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7/23/2019 Formulario das Rendas
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FORMULRIODAS
RENDAS
1. RENDAS CERTAS TEMPORRIAS DE TERMOS CONSTANTES, EM TEMPO DISCRETO E EM TEMPO CONTNUO
2. RENDAS CERTAS PERPTUAS DE TERMOS CONSTANTES, EM TEMPO DISCRETO E EM TEMPO CONTNUO
3. RENDAS CERTAS TEMPORRIAS, DE TERMOS A VARIAR EM PROGRESSO GEOMTRICA, EM TEMPO DISCRETO
4. RENDAS CERTAS PERPTUAS, DE TERMOS A VARIAR EM PROGRESSO GEOMTRICA, EM TEMPO DISCRETO5. RENDAS CERTAS TEMPORRIAS, DE TERMOS A VARIAR EM PROGRESSO ARITMTICA, EM TEMPO DISCRETO
6. RENDAS CERTAS PERPTUAS, DE TERMOS A VARIAR EM PROGRESSO ARITMTICA, EM TEMPO DISCRETO
7. RENDAS CERTAS COM TERMOS A VARIAR DE OUTROS MODOS, EM TEMPO DISCRETO
. RENDAS CERTAS COM TERMOS VARIVEIS, EM TEMPO CONTNUO
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1. RENDA CERTA TEMPORRIA DE TERMOS CONSTANTES ! n "#$%&' ()*"+$*&' & -#$&/& /0 $#)/0 &*)*/# &% & -#$&/& /0 "00 i
I)"#*$0
I%#/*0"0
T#$%&' )&$%0*'
1 nn i
v
ia =
( ) ( ) 1 11 n
n
n i n i
ii
is a+ = +=
T#$%&' 0)"#*-0/&'( ) 1 11n i n in i ia a a = += +&&
( ) ( )1
1 1
n n
n i n i n ii is a a
++ = +=&& &&
D*#$*/0 k -#$&/&' /0"00 i
T#$%&' )&$%0*'
k
n i n ik va a=
( ) ( ) ( )1 1n k n
n i n i n i n ik ki is a a s
++ = + ==
T#$%&' 0)"#*-0/&'
1
k k
n i n i n ik v va a a
==&& &&
( ) ( ) ( )1
1 1
n k n
n i n i n i n ik ki is a a s
+ ++ = + ==&& && &&
F$0*&)0/0
!& "#$%& ()*"+$*&
-0& 0& &)&/& -#$&/& /0$#)/0, #% m-$#'"089#'#:(*/*'"0)"#' /#;0&$ 1< m
I%#/*0"0
T#$%&')&$%0*'
( )
( ) m
n i n im
ia a
i
=
( )( )
( )( )
( ) ( )1 1 =
n nm m
n i n i n i n im mi i
i is a a s
i i
+ = +
=
T#$%&'0)"#*-0/&'
( )( )
( )( )
( )1 1
1 1m m
n i n i n i
m m
mi i
ia a a
i
== + +&&
( )( )
( )( )
( ) ( )
( )1 11
1 =1 1nm m
n i n i n i n i
m mm
m mii i
i is s a s
i i
+ = +
= + +&&
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!>F$0*&)0/0
!& "#$%& ()*"+$*&
-0& 0& &)&/& -#$&/& /0$#)/0, #% m-$#'"089#'#:(*/*'"0)"#' /#;0&$ 1< m
D*#$*/0k
-#$&/&' /0"00 i
T#$%&')&$%0*'
( )
( ) m
n i n i
k
mk v
ia a
i
=
( )( )
( )( )
( )( )
( )
( )
( )
1 1 1 =m m m
n i n i n i n i n i
mn k n n
mk ki i i
is a a a s
i
++ = + = +=
T#$%&'0)"#*-0/&'
( )( )
( )
( )1
11
m m m
n i n i n i
m
mk k
ki v
ia a a
i
+ ==&&
( )( )
( )( )
( )( )
( )
( )( )1 1
1 1 1m m m m
n i n i n i n i n i
n k n m n m
mk ki i i
is a a a s
i
+ + ++ = + = + ==&& && &&
C&)")(0!& "#$%& ()*"+$*& -0& /# &$%0&)")(0 0& &)& /&-#$&/& /0 $#)/0, 0(%0 "00 "0%?%()*"+$*0
I%#/*0"0T#$%&' )&$%0*'T#$%&' 0)"#*-0/&'
@
n
t
n i n i
iv dta a
== ( ) ( ) ( )
@
1 1 1 =
n
n n t n
n i n i n i n i
i ii i dt is a a s
+ = + = +
=
D*#$*/0 k -#$&/&' /0"00 i
T#$%&' )&$%0*'T#$%&' 0)"#*-0/&'
k k
n i n i n ik
iv va a a
==
( ) ( ) ( ) ( )1 1 1n k n n
n i n i n i n i n ik
ii i is a a a s
++ = + = + ==
N&"0' S# &' n "#$%&', #%?&$0 &)'"0)"#', &$#% /# (% :(0:(#$ ;0&$ @T> , ?0'"0 %("*-*0$ &' '%?&&' # 0' #-$#''9#' 0)"#$*&$#' -&$ T , -0$0'# &?"#$#% &' ;0&$#' 0"(0*' # 0(%(0/&' /#''0' $#)/0'.
C&%& &* ;*'"& )(%0 /0' 0(0', '#%-$# -&'';# &);#$"#$ (%0 $#)/0 $0*&)0/0 )(%0 $#)/0 *)"#*$0 &% *(0 /*0$0%0 $&)&B*&. P0$0*''&, ?0'"0$+ -$/#$ %(/0)80 /& -#$&/& /0 $#)/0 # 0"#)/#$ ' 0"#$089#' :(# /0 /#&$$#% #% "#$%&' /0 "00 /# ($& 0 ('0$, /& ;0&$ /#0/0 "#$%& # /& )%#$& /#'"#'.
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2. RENDA CERTA PERPTUA DE TERMOS CONSTANTES!*)*)*"&' "#$%&' ()*"+$*&' & -#$&/& /0 $#)/0 &*)*/# &% & -#$&/& /0 "00 i
S# 0' $#)/0' '& -#$-#"(*/0/#', &' $#'-#"*;&' ;0&$#' 0"(0*' # 0(%(0/&' &?"%H'# "&%0)/& & *%*"# /0' #-$#''9#' 0*%0, &% n . D0$#'("0 :(# "&/&' &' ;0&$#' 0(%(0/&' . O' ;0&$#' 0"(0*' ;% *)/*0/&' )0' "$' "0?#0' '#(*)"#'.
I)"#*$0
I%#/*0"0
T#$%&' )&$%0*' 1
i ia =
T#$%&' 0)"#*-0/&' 1
i
i
ia
+=&&
D*#$*/0 k -#$&/&' /0"00 i
T#$%&' )&$%0*'
k
ik i
va =
T#$%&' 0)"#*-0/&'1
k
ik i
va
=&&
F$0*&)0/0
I%#/*0"0
T#$%&' )&$%0*' ( )
( )
1mi m
a
i
=
T#$%&' 0)"#*-0/&' ( ) ( )
( )
11m
i
m
m
ia
i
+=&&
D*#$*/0 k -#$&/&' /0"00 i
T#$%&' )&$%0*' ( )
( )
m
i
k
mk
va
i
=
T#$%&' 0)"#*-0/&' ( )
( )
1m
i
k m
mk
va
i
=&&
C&)")(0I%#/*0"0 T#$%&' )&$%0*'
T#$%&' 0)"#*-0/&'@
1t
i v dta
== D*#$*/0 k -#$&/&' /0"00 i
T#$%&' )&$%0*'T#$%&' 0)"#*-0/&'
k
ik
va
=
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3. RENDA CERTA TEMPORRIA, DE TERMOS A VARIAR EM PROGRESSO GEOMTRICA ! n "#$%&' 0 ;0$*0$ #% -$&$#''&%"$*0 /# $0& 1 r+ , '#)/& & -$*%#*$& "#$%& ()*"+$*& & -#$&/& /0 $#)/0 &*)*/# &% & -#$&/& /0 "00 i
I)"#*$0
I%#/*0"0
T#$%&')&$%0*'
( ) ( )21 J
1 , J ,V07&$ 0"(07 1 1 1 1
,
nn n ii ri i r
r r r r
nv i r
av v v
= + + + + + +
=
== + L
( )
( )
( )
J
1
1, J ,
V0&$ A(%(0/& V0&$ 0"(0 1 1 1
1 ,
n
n
n i
n
i i ri i r
i r r
n i i r
a
+ =
= + + + + =
=
T#$%&'0)"#*-0/&'
( ) ( )11 J
1, J ,
V0&$ 0"(0 1 1 1 1
,
1 nn n i
i i ri i r
r r r r
n i r
av v
+ = + + + + + +
=
== + L
( )( )
( )
1
J
1, J ,
V0&$ A(%(0/& V0&$ 0"(0 1 1 1
1 ,
n
n
n i
n
i i ri i r
i r r
n i i r
a
+ + = = + + +
+ =
=
D*#$*/0 k -#$&/&' /0"00 i
T#$%&')&$%0*'
( ) ( )1
1
1 2 J, J ,
V0&$ 0"(0 1 1 1 1
,
k
n
k
k k k n n i
v i ri i r
r r r r
nv i r
av v v
+
+ + +
= + + + + + + =
== + L
( )
( )
( )
J
1
1, J ,V0&$ A(%(0/& V0&$ 0"(0 1 1 1
1 ,
n k
n
n i
n
i i r
i i ri r r
n i i r
a+
+ = = + + +
+ =
=
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I)"#*$0D*#$*/0 k -#$&/&' /0"00 i
T#$%&'0)"#*-0/&'
( ) ( )
1
11 1
1
J , J ,V0&$ 0"(0 1 1 1 1
,
k
nk k k n
k
n i
i r
i i rr r r r
n i r
v
av v v
v
+ +
= + + + + + + =
== + L
( )( )
( )
1
J
1, J ,
V0&$ A(%(0/& V0&$ 0"(0 1 1 1
1 ,
n k
n
n i
n
i i ri i r
i r r
n i i r
a+
+ + = = + + +
+ =
=
D#;# )&"0$H'# :(# 0' #-$#''9#' 0*%0 &)"*)(0% ;+*/0' %#'%& :(0)/& J @.i r i< < N#'"# 0'&, # 0& &)"$+$*& /& :(# K0?*"(0, &' ;0&$#'0"(0*' ;% '(-#$*&$#' 0 n # "0)"& %0*' '(-#$*&$#' :(0)"& %0*' ##/#$ .r i .
A' $#)/0' $0*&)0/0' "% "$0"0%#)"& '#%#K0)"# 0& :(# '# ;*( 0"$+', %0' & %#K&$ %#'%& &);#$"H0' #% $#)/0' *)"#*$0', 0*)/&H'# )&'0'&' -$##/#)"#'. (0)"& ' $#)/0' &)")(0' /# "#$%&' ;0$*+;#*', '#$& &?#"& /# (%0 0?&$/0#% #)$*0 )0 -0$"# *)0 /0 #-&'*8&.
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4. RENDA CERTA PERPTUA, DE TERMOS A VARIAR EM PROGRESSO GEOMTRICA !*)*)*"&' "#$%&' 0 ;0$*0$ #%-$&$#''& #&%"$*0 /# $0& 1 r+ , '#)/& & -$*%#*$& "#$%& ()*"+$*& & -#$&/& /0 $#)/0 &*)*/# &% & -#$&/& /0 "00 i .
T0 &%& 0)"#$*&$%#)"#, :(0)/& 0' $#)/0' '& -#$-#"(*/0/#', &' $#'-#"*;&' ;0&$#' 0"(0*' # 0(%(0/&' &?"%H'# "&%0)/& & *%*"# /0'#-$#''9#' 0*%0, &% n . D0 $#'("0 :(# "&/&' &' ;0&$#' 0(%(0/&' !# 0" 0()' /&' ;0&$#' 0"(0*', -&*' 0 '$*#( )
( )
1 1
1 1
1 1
1 11
n n
n
n n
i r r
i ii
= =
+ + = + + + )& &);#$#)"# :(0)/& )& '# ;#$**0 0 &)/*8& 11 11
r
i
+ <
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5. RENDA CERTA TEMPORRIA, DE TERMOS A VARIAR EM PROGRESSO ARITMTICA ! n "#$%&' 0 ;0$*0$ #% -$&$#''&0$*"%"*0 /# $0& h , '#)/& & -$*%#*$& "#$%& /# ;0&$ C & -#$&/& /0 $#)/0 &*)*/# &% & -#$&/& /0 "00 i . E% -$*)-*&, #%?&$0 -&''0 "#$H'#
@,h< )& 0 '#)"*/& 0/%*"*$ :(# 0 -0$"*$ /# #$"0 &$/#% &' "#$%&' ;% )#0"*;&'.
C&%#80)/& -&$ &)'*/#$0$ & 0'& -0$"*(0$ #% :(# 1h C= = , /#/(H'# :(# & ;0&$ 0"(0 /# (%0 "0 $#)/0
( ) 2 3 2 3 n in i
n
n n
Ia v ni
a vv v v
= + + + + =L
& & '#( ;0&$ 0(%(0/& ( ) ( ) ( ) 1 n i
n i n i
n ni
iIs Ia
s = + = . E)"&
I)"#*$0
I%#/*0"0
T#$%&')&$%0*'
( ) ( )( ) ( ) ( )2 V0&$ 0"(0 1 n i n in
C h C n h C h h IaCv v v a+ + + + +== + L
( ) ( ) ( )
V0&$ 0(%(0/& V0&$ 0"(0 1n i n i
ni C h h Iss+ +==
T#$%&'0)"#*-0/&'
( ) ( ) ( )V0&$ 0"(0 1n i n iC h i h Iaa + += &&
( ) ( ) ( ) ( )
V0&$ 0(%(0/& V0&$ 0"(0 1 1n i n i
ni C h i h Iss+ + +== &&
D*#$*/0 k -#$&/&' /0"00 i
T#$%&')&$%0*'
( ) ( )V0&$ 0"(0 n i n ik
v C h h Iaa = + ( ) ( ) ( )V0&$ 0(%(0/& V0&$ 0"(0 1
n k
n i n ii C h h Iss
++ +==
T#$%&'0)"#*-0/&'
( ) ( ) ( )
V0&$ 0"(0 1kn i n i
C h i h Iav a + + = &&
( ) ( ) ( ) ( )
V0&$ 0(%(0/& V0&$ 0"(0 1 1n i n i
n ki C h i h Iss
++ + +== &&
A )&"0 0)"#$*&$, '&?$# 0' $#)/0' $0*&)0/0', "0%?% ;+*/0 0&$0.U% &("$& 0'& -0$"*(0$ *)"#$#''0)"# 0 '*"(08& #% :(# 1,h C n= = . O ;0&$ 0"(0 /0 $#)/0 &$$#'-&)/#)"#
( ) ( ) ( ) ( ) ( )2 3
1 2 1n in i n i
nDa nv n n n Iav v v a= + + + + +
=L # & '#( ;0&$ 0(%(0/&
( ) ( )
( ) ( ) ( )
1 1n i n in i n i
ni n s IsDs Da= + = + .
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6. RENDA CERTA PERPTUA, DE TERMOS A VARIAR EM PROGRESSO ARITMTICA !*)*)*"&' "#$%&' 0 ;0$*0$ #% -$&$#''&0$*"%"*0 /# $0& h , '#)/& & -$*%#*$& "#$%& /# ;0&$ C & -#$&/& /0 $#)/0 &*)*/# &% & -#$&/& /0 "00 i
R#-#"*)/& & -$''& + K0?*"(0, "&%#H'# & *%*"# /# 0/0 (%0 /0' #-$#''9#' 0*%0, &% n . C&%#80)/& )&;0%#)"# -&$ &)'*/#$0$ & 0'&
-0$"*(0$ #% :(# 1h C= = , /#/(H'# :(# & ;0&$ 0"(0 /# (%0 "0 -#$-#"(*/0/#
( ) ( )2 32
12 3 .*% *%n n
n i
i n i
n
n n i
Ia v n Iai i
a vv v v
+= + + + + + = ==L L
&&
O' $#'"0)"#' ;0&$#' 0"(0*' ;% *)/*0/&' )0 "0?#0 '#(*)"# !&' ;0&$#' 0(%(0/&' .
I)"#*$0
I%#/*0"0
T#$%&')&$%0*' 2
V0&$ 0"(0 Ci h
i
+=
T#$%&'0)"#*-0/&'
( ) ( )2
1
V0&$ 0"(0
Ci h i
i
+ +
=
D*#$*/0 k -#$&/&' /0"00 i
T#$%&')&$%0*'
( )2
V0&$ 0"(0k
v Ci h
i
+=
T#$%&'0)"#*-0/&'
( )1
2V0&$ 0"(0
kv Ci h
i
+=
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7. RENDA CERTA COM TERMOS A VARIAR DE OUTROS MODOS!#% "#%-& /*'$#"&
SB '# 0-$#'#)"0 & 0'& /0' $#)/0' "#%-&$+$*0', *%#/*0"0' # *)"#*$0', /# "#$%&' )&$%0*', '#0% 1 2, , , nL L LL , -&*' 0 0/0-"08& ' &("$0' '*"(089#'
*%#/*0"0.
V#%
V0&$ 0"(01
nt
tt
v L
==
V0&$ 0(%(0/& ( ) ( ) ( )1
V0&$ A"(0 1 1n
n n t
tt
i i L
== + = +
. RENDA CERTA COM TERMOS VARIVEIS !#% "#%-& &)")(&
(0)/& 0 $#)/0 -00 &)"*)(0%#)"# )& *)"#$;0& [ ]@, n # & -00%#)"& '# 0 0 (%0 "00 ( )h t , :(# ()8& &)")(0 /& "#%-&, *%#/*0"& :(#
V0&$ 0"(0 ( )@
n
tv h t d t =
V0&$ 0(%(0/& ( ) ( ) ( ) ( )@
V0&$ A"(0 1 1nn
n t
i i h t dt
+ == +
.
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N& 0'& -0$"*(0$ #% :(# ( )h t t= , # -#0 B?;*0 0)0&*0 &% & 0'& /*'$#"&, "#%H'#
V0&$ 0"(0 ( )@
n
t n i
n i
nn
Ia v tdt va
=
= =
V0&$ 0(%(0/& ( ) ( ) ( ) V0&$ A"(0 1* n n in i
nIs
s
==
= .
M0' '# &$ ( )h t n t = , + '# ;0* "#$
V0&$ 0"(0 ( ) ( )@
n
t n i
n ivDa n t dt
n a
= ==
V0&$ 0(%(0/& ( ) ( ) ( )
V07&$ A"(07 1O*1
n n i
n i
nn i
Dss
==
+
= .
T&/0' 0' /#/(89#' 0)"#$*&$#' &$0% #*"0' )& -$#''(-&'"& /& RC !&$80 /# ($& &)'"0)"# 0& &)& /& "#%-& ( )( )) 1t i = + . (0)/& )& *''&:(# 0&)"##, # '# "#% t ()8& )& &)'"0)"# /& "#%-&, *%-9#%H'# 0' &$$#'-&)/#)"#' 0/0-"089#'. P&$ ##%-&, )& 0'& /0 $#)/0 &)")(0;#%
V0&$ 0"(0 ( )@
@
n
t
sds
h t e dt
=
V0&$ 0(%(0/& ( ) ( ) ( )@ @
@ @ @V0&$ 0"(0
n n
n t n n
s s s s
t
ds ds ds ds
e h t e dt e h t e dt
=
= =
.
M0' "(/& '# -0''0 /# *(0 %&/&, '# &$ (%0 $#)/0 #% "#%-& /*'$#"&. A-#)0' '# 0 0 0"(0*08& !&( 0 0-*"0*08& /# 0/0 "#$%&$#&$$#)/& &$80 /# ($& &%&, 0*)0, '# # "0%?% )& RC, &)/# "(/& *0 %0*' '*%-#' 'B -#& 0"& /# 0:(#0 '#$ &)'"0)"#.
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