formulario das rendas

7/23/2019 Formulario das Rendas

1/11

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

1


2/11

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

+ = +

= + +&&

2


3/11

!>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 "#$%& # /& )%#$& /#'"#'.

3


4/11

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

=

4


5/11

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+

+ = = + + +

+ =

=

5


6/11

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&.

6


7/11

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

+ <


8/11

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= + = + .


9/11

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

+=


10/11

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

+ == +

.

1@


11/11

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)"#.

11

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