Question

Quantos termos tem a P.A ( 5x+1, 6x, 7x-1 , ... , 13x-7 )

Answer 1

$\mathrm{PA=(5x+1, 6x, 7x-1,\dots,13x-7)\ \ \| \ \ n=\ ?}\\\\ \textbf{Calculando a raz\~ao da PA:}\\\\ \mathrm{r=a_2-a_1\ \to\ r=6x-(5x+1)}\\ \mathrm{r=6x-5x-1\ \to\ \boxed{\mathrm{r=x-1}}}\\\\ \textbf{Calculando n atrav\'es do termo geral:}\\\\ \mathrm{a_n=a_k+(n-k).r\ \to\ a_n=a_1+(n-1).r}\\ \mathrm{13x-7=5x+1+(n-1).(x-1)}\\ \mathrm{13x-7-5x-1=(n-1).(x-1)}\\ \mathrm{(n-1).(x-1)=8x-8}\\ \mathrm{(n-1).(x-1)=8.(x-1)}\\ \mathrm{n-1=8\ \to\ n=8+1\ \to\ \boxed{\mathrm{n=9}}}$

$\mathbf{Resposta\ \to\ \boxed{\boxed{\mathbf{n=9\ termos}}}}$