Question

Olá pode me ajudar nessa questão ??
Calcule a soma dos cinquenta primeiros termos da PA (1/2, 2, 7/2, 4,...)

Answer 1

1° que saber qual é a razão, e qual é o último termo.


Calculando a razão:

$PA~\left( ~\dfrac{1}{2} ,2, \dfrac{7}{2} , 4, ...~ \right)\\\\\\\\\\ r=a_2-a_1\to~~ r=2- \dfrac{1}{2}\to~~ r=\dfrac{4-1}{2}\to~~ \large\boxed{ r=\dfrac{3}{2}}$




Calculando o último termo:


$a_1= \frac{1}{2} \\r=\frac{3}{2}\\ n=50\\ a_{50}=?\\\\\\ a_n=a_1+(n-1)\times r \\\\\\ a_{50}= \dfrac{1}{2} +(50-1) \times \dfrac{3}{2}\to\\\\ a_{50}= \dfrac{1}{2} +49 \times \dfrac{3}{2}\to \\\\ a_{50}= \dfrac{1}{2} + \dfrac{147}{2}\to \\\\ a_{50}= \dfrac{1+147}{2} \to\\\\ a_{50}= \dfrac{148}{2}\\\\ \large\boxed{ a_{50}= 74 }$




Agora sim, passamos ao cálculo da soma de termos:


$S_n= \dfrac{\left( a_1+a_n\right) \times n}{2} \\\\\\\\ S_{50}= \dfrac{\left( \dfrac{1}{2}+74 \right) \times 50}{2}\to ~~~~ S_{50}= \dfrac{\left( \dfrac{1+148}{2} \right) \times 50}{2}\to \\\\\\ S_{50}= \dfrac{\left( \dfrac{149}{2} \right) \times 50}{2}\to ~~~~ S_{50}= \dfrac{ \dfrac{7450}{2}}{2}\to ~~~~S_{50}= \dfrac{7450}{2}\times \dfrac{1}{2} \to \\\\\\S_{50}= \dfrac{7450}{4} \to~~~~ \Large\boxed{\boxed{S_{50}= \dfrac{3725}{2}} }$