Question

Determine o primeiro termo e a razão da progressão aritmética sabendo que a5 + a6 = 73 e a2 + a8 = 66

Answer 1

Dados:

 $\displaystyle \mathsf{a_5 + a_6 = 73}\\ \displaystyle \mathsf{a_2 + a_8 = 66}\\ \displaystyle \mathsf{a_1 = ?}\\ \displaystyle \mathsf{r = ?}$

Cálculo:

$\displaystyle \mathsf{a_5 + a_6 = 73}\\ \\ \displaystyle \mathsf{a_5 = a_1 + 4r}\\ \displaystyle \mathsf{a_6 = a_1 + 5r}\\ \\ \displaystyle \mathsf{(a_1 + 4r) + (a_1 + 5r) = 73}\\ \displaystyle \mathsf{a_1 + 4r + a_1 + 5r = 73}\\ \displaystyle \boxed{\mathsf{2a_1 + 9r = 73}}$

$\displaystyle \mathsf{a_2 + a_8 = 66}\\ \\ \displaystyle \mathsf{a_2 = a_1 + r}\\ \displaystyle \mathsf{a_8 = a_1 + 7r}\\ \\ \displaystyle \mathsf{(a_1 + r) + (a_1 + 7r) = 66}\\ \displaystyle \mathsf{a_1 + r + a_1 + 7r = 66}\\ \displaystyle \boxed{\mathsf{2a_1 + 8r = 66}}$

$\displaystyle \mathsf{ \left \{ {{2a_1 + 9r = 73} \atop {2a_1 + 8r = 66 \: (-1)}} \right. }\\ \\ \\ \displaystyle \mathsf{ \left \{ {{2a_1 + 9r = 73} \atop {-2a_1 - 8r = -66}} \right. }\\ \displaystyle \mathsf{\rule{90}{0,7}}\\ \displaystyle \boxed{\mathsf{r = 7}}$

$\displaystyle \mathsf{2a_1 + 8r = 66}\\ \\ \displaystyle \mathsf{r = 7}\\ \\ \displaystyle \mathsf{2a_1 + 8.7 = 66}\\ \displaystyle \mathsf{2a_1 + 56 = 66}\\ \displaystyle \mathsf{2a_1 = 66 - 56}\\ \displaystyle \mathsf{2a_1 = 10}\\ \\ \displaystyle \mathsf{a_1 = \frac{10}{2}}\\ \\ \displaystyle \boxed{\mathsf{a _1 = 5}}$

Resposta: O primeiro termo é 5 e a razão, 7.