Question

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

Answer 1

$a_3=10$

$a_{98}=90$

$a_n=a_1+(n-1)r$

$a_3=a_1+2r$

$a_{98}=a_1+97r$

$a_1+2r=10~(i)$

$a_1+97r=90~(ii)$

Fazendo $(ii)-(i)$:

$(a_1+97r)+(a_1+2r)=90-10$

$95r=80$

$r=\dfrac{80}{95}$

$r=\dfrac{16}{19}$

$a_1+2r=10$

$a_1+\dfrac{32}{19}=10$

$a_1=10-\dfarc{32}{19}$

$a_1=\dfrac{158}{19}$


$a_{100}=\dfrac{158}{19}+99\cdot\dfrac{16}{19}$

$a_{100}=\dfrac{1~742}{19}$


$S_{100}=\dfrac{(\frac{158}{19}+\frac{1~742}{19})100}{2}=\frac{1~900}{19}\cdot50=100\cdot50=5~000$