Question

calcule a soma das 24 primeiros termos de cada PA

a.(-57, -27, 3...)

b.(\frac{2}{3}, \frac{8}{3}, \frac{14}{3}...)

c.(7, 7, 7...)

d.(-\frac{1}{2},-\frac{1}{4}, 0...)

Answer 1

Soma de uma P.A. tem como fórmula:

$<var>\boxed{Sn = \frac{(a1 + an)*n}{2}}</var>$

 

Primeiramente vamos calcular a razão de cada P.A.

a) $<var>-27-(-57) = -27 + 57 = \boxed{30}</var>$

 

b) $<var>\frac{8}{3} - \frac{2}{3} = \frac{6}{3} = \boxed{2}</var>$

 

c) $<var>7 - 7 = \boxed{0}</var>$

 

d) $<var>-\frac{1}{4} - (-\frac{1}{2}) = -\frac{1}{4} + \frac{1}{2} = -\frac{1}{4} + \frac{2}{4} = \boxed{\frac{1}{4}}</var>$

 

Definida as razão, vamos calcular a soma dos 24 primeiros termos.

a) $<var>Sn = \frac{(a1 + an)*n}{2} </var>$

 

$<var>S24 = \frac{(-57 + a24)*24}{2}</var>$

 

Vamos descobrir o a24 para podermos substituir:

$<var>an = a1 + (N-1)*r</var>$

 

$<var>a24 = -57 + (24-1) *30</var>$

 

$<var>a24 = -57 + 23*30</var>$

 

$<var>a24 = -57 + 690</var>$

 

$<var>a24 = 633</var>$

 

Voltando na fórmula:

$<var>S24 = \frac{(-57 + 633)*24}{2} </var>$

 

$<var>S24 = \frac{(576)*24}{2}</var>$

 

$<var>S24 = \frac{13824}{2}</var>$

 

$<var>\boxed{S24 = 6912}</var>$

 

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b) $<var>S24 = \frac{(\frac{2}{3} + a24)*2}{2}</var>$

 

$<var>a24 = \frac{2}{3} + (24-1) * 2</var>$

 

$<var>a24 = \frac{2}{3} + 23 * 2</var>$

 

$<var>a24 = \frac{2}{3} + 46</var>$

 

$<var>a24 = \frac{2}{3} + \frac{138}{3} = \frac{140}{3}</var>$

 

 

$<var>S24 = \frac{(\frac{2}{3} + \frac{140}{3})*2}{2} </var>$

 

Anula os dois.

$<var>S24 = \frac{2}{3} + \frac{140}{3} = \boxed{\frac{142}{3}}</var>$

 

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c) A soma dos primeiros 24 termos é 24*7

$<var>S24 = 7*24</var>$

 

$<var>\boxed{S24 = 168}</var>$

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d) $<var>Sn = \frac{(a1 + an)*R}{2}</var>$

 

$<var>S24 = \frac{(-\frac{1}{2} + a24)*\frac{1}{4}}{2}</var>$

 

Vamos calcular o termo a24

$<var>an = a1 + (n-1) * R</var>$

 

$<var>a24 = -\frac{1}{2} + (24-1) * \frac{1}{4}</var>$

 

$<var>a24 = -\frac{1}{2} + 23 * \frac{1}{4}</var>$

 

$<var>a24 = -\frac{1}{2} + \frac{23}{4}</var>$

 

$<var>a24 = -\frac{2}{4} + \frac{23}{4} = \frac{21}{4}</var>$

 

 

$<var>S24 = \frac{(-\frac{1}{2} + \frac{21}{4})*\frac{1}{4}}{2}</var>$

 

$<var>S24 = \frac{(-\frac{2}{4} + \frac{21}{4})*\frac{1}{4}}{2}</var>$

 

$<var>S24 = \frac{\frac{19}{4}*\frac{1}{4}}{2}</var>$

 

$<var>S24 = \frac{\frac{19}{16}}{2}</var>$

 

$<var>\boxed{S24 = {\frac{19}{32}}}</var>$

Answer 2

Resposta:

Explicação passo-a-passo:

a)

an = -57 + (24 - 1) * 30

an = -57 + 690

an = 633

 

b)

an = 2/3 + (24-1) * 2

an = 2/3 + 46

an = 140 / 3

 

c) PA constante. 24ª termo = 7

 

d) an = -1/2 + (24-1). (-1/4)

an = -1/2 - 23/4

an= -50/8  

 

CALCULANDO A SOMA DOS TERMOS.

A)

S24 = [(-57 + 633) * 24]/2

= 576 * 12 = 6912

 

B)

s24 = [ (140/3 + 2/3) * 24]/2

= 142/3 * 12 = 1704 /3 = 568.

 

c) S24 = [(7+7)*24]/2 = 14 * 24/2 = 336/2 = 168

 

d) S24=[(-1/2 - 50/8) * 24] /2    *** -1/2 = -4/8

= -54/8 * 12 = -648 / 8 = -81