Question

Calcule a soma dos 24 primeiros termos de cada PA
a)(-57,-27,3,...)
b)(2/3,8/3,14/3,...)
c)(7,7,7,...)
d)(-1/2,-1/4,0,...)

Answer 1

$\large a) ~A~ soma ~dos ~24 ~primeiros ~termos ~da ~PA ~ \Rightarrow ~ S24 = 6912$

$\large b) ~A~ soma ~dos ~24 ~primeiros ~termos ~da ~PA ~ \Rightarrow ~ S24 = 568$

$\large c) ~A~ soma ~dos ~24 ~primeiros ~termos ~da ~PA ~ \Rightarrow ~ S24 = 168$

$\large d) ~A~ soma ~dos ~24 ~primeiros ~termos ~da ~PA ~ \Rightarrow ~ S24 = 57$

                                  $\Large Progress\tilde{a}o ~Aritm\acute{e}tica$

a)

Encontrar a razão da PA:

$r = a2 - a1\\\\r = -27 - ( - 57)\\\\r = -27 + 57\\\\r = 30$

Encontrar o valor do termo a24

$an = a1 + ( n -1 ) . r\\\\a24 = -57 + ( 24 -1 ) . 30\\\\a24 = -57 + 23 . 30\\\\a24 = -57 + 690\\\\a24 = 633$

Soma dos termos:

$Sn = ( a1 + an ) ~. ~n ~/~ 2\\\\ S24 = ( -57 + 633 )~ . ~24~ /~ 2 \\\\ S24 = 576~ . ~12\\\\ S24 = 6912$  

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b)

Encontrar a razão da PA:

$r = a2 - a1\\\\r = \dfrac{8}{3} - \dfrac{2}{3}\\\\\\ r = 2$

Encontrar o valor do termo a24:

$an = a1 + ( n -1 ) . r\\\\\\a24 = \dfrac{2}{3} + ( 24 -1 ) ~. ~2\\\\\\ a24 = \dfrac{2}{3} + 23~ . ~2\\\\\\ a24 = \dfrac{2}{3} + 46\\\\\\a24 = \dfrac{140}{3}$  

Soma dos termos:

$Sn = ( a1 + an ) . n / 2\\\\\\ S24 = ( \dfrac{2}{3} + \dfrac{140}{3} )~ . ~24~ /~ 2 \\\\\\ S24 = \dfrac{142}{3}~ .~ 12\\\\\\ S24 = 568$

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c)

Encontrar a razão da PA:

$r = a2 - a1\\\\r = 7 - 7\\\\r = 0$

Encontrar o valor do termo a24:

$an = a1 + ( n -1 ) . r\\\\a24 = 7 + ( 24 -1 ) . 0\\\\a24 = 7 + 23 . 0\\\\a24 = 7 + 0\\\\a24 =7$

Soma dos termos

$Sn = ( a1 + an )~ . ~n~ /~ 2\\\\ S24 = ( 7 + 7 )~ .~ 24~ /~ 2\\\\ S24 = 14 ~.~ 12\\\\ S24 = 168$  

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d)

Encontrar a razão da PA:

$r = a2 - a1\\\\r = -\dfrac{1}{4} - ( -\dfrac{1}{2} )\\\\\\r = -\dfrac{1}{4} + \dfrac{1}{2}\\\\\\r = \dfrac{1}{4}$

Encontrar o valor do termo a24:

$an = a1 + ( n -1 ) . r\\\\\\a24 = -\dfrac{1}{2} + ( 24 -1 )~ .~ \dfrac{1}{4} \\\\\\ a24 = -\dfrac{1}{2} + 23 ~.~\dfrac{1}{4} \\\\\\ a24 = -\dfrac{1}{2} + \dfrac{23}{4} \\\\\\ a24 = \dfrac{21}{4}$  

Soma dos termos:

$Sn = ( a1 + an )~ .~ n~ / ~ 2\\\\\\ S24 = ( -\dfrac{1}{2} + \dfrac{21}{4} ) ~. ~24~ /~ 2 \\\\\\ S24 = \dfrac{19}{4}~ . ~12\\\\\\ S24 = 57$  

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