Matemática
paulovitor000
74

1.      (MACK – SP)  Numa PA de 100 termos,  a3 = 10 e a98 = 90. A soma de todos os termos é:

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(1) Respostas
Jailson123

[latex]a_3=10[/latex] [latex]a_{98}=90[/latex] [latex]a_n=a_1+(n-1)r[/latex] [latex]a_3=a_1+2r[/latex] [latex]a_{98}=a_1+97r[/latex] [latex]a_1+2r=10~(i)[/latex] [latex]a_1+97r=90~(ii)[/latex] Fazendo [latex](ii)-(i)[/latex]: [latex](a_1+97r)+(a_1+2r)=90-10[/latex] [latex]95r=80[/latex] [latex]r=\dfrac{80}{95}[/latex] [latex]r=\dfrac{16}{19}[/latex] [latex]a_1+2r=10[/latex] [latex]a_1+\dfrac{32}{19}=10[/latex] [latex]a_1=10-\dfarc{32}{19}[/latex] [latex]a_1=\dfrac{158}{19}[/latex] [latex]a_{100}=\dfrac{158}{19}+99\cdot\dfrac{16}{19}[/latex] [latex]a_{100}=\dfrac{1~742}{19}[/latex] [latex]S_{100}=\dfrac{(\frac{158}{19}+\frac{1~742}{19})100}{2}=\frac{1~900}{19}\cdot50=100\cdot50=5~000[/latex]

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