Tìm 4 số lập thành một CSN biết tổng bốn số bằng 15 và tổng các bình phương của chúng bằng 85

Trả lời

Giả sử 4 số cần tìm là ${\text{a; aq; aq}}^{\text{2}}{\text{; aq}}^{\text{3}}\text{, a}\ne \text{0}$

Theo đề bài ta có: $\left\{\begin{array}{c}{\text{a + aq + aq}}^{\text{2}}{\text{ + aq}}^{\text{3}}\text{ = 15}\\ {\text{a}}^{\text{2}}{\text{ + a}}^{\text{2}}{\text{q}}^{\text{2}}{\text{ + a}}^{\text{2}}{\text{q}}^{\text{4}}{\text{ + a}}^{\text{2}}{\text{q}}^{\text{6}}\text{ = 85}\end{array}\right.$

$\iff \left\{\begin{array}{c}{\text{a(1 + q + q}}^{\text{2}}{\text{ + q}}^{\text{3}}\text{) = 15}\\ {\text{a}}^{\text{2}}{\text{(1 + q}}^{\text{2}}{\text{ + q}}^{\text{4}}{\text{ + q}}^{\text{6}}\text{) = 85}\end{array}\right.$

$\left\{\begin{array}{c}{\text{a}}^2{{\text{(1 + q + q}}^{\text{2}}{\text{ + q}}^{\text{3}}\text{)}}^2{\text{ = 15}}^2\\ {\text{a}}^{\text{2}}{\text{(1 + q}}^{\text{2}}{\text{ + q}}^{\text{4}}{\text{ + q}}^{\text{6}}\text{) = 85}\end{array}\right.$

$\iff \frac{{{\text{(1 + q + q}}^{\text{2}}{\text{ + q}}^{\text{3}}\text{)}}^{\text{2}}}{{\text{(1 + q}}^{\text{2}}{\text{ + q}}^{\text{4}}{\text{ + q}}^{\text{6}}\text{)}}\text{ = }\frac{{\text{15}}^{\text{2}}}{\text{85}}$

$\iff {\text{14q}}^{\text{6}}{\text{ - 17q}}^{\text{5}}{\text{ - 3q}}^{\text{4}}{\text{ - 34q}}^{\text{3}}{\text{ - 3q}}^{\text{2}}\text{ - 17q + 14 = 0}$

$\iff {\text{(q - 2)(2q - 1)(7q}}^{\text{2}}{\text{ + q + 7)(q}}^{\text{2}}\text{ + 1) = 0}$

$\iff \left[\begin{array}{c}q=2\\ q=\frac{1}{2}\end{array}\right.$

Với $\text{q = 2}\Rightarrow \text{a = }\frac{\text{15}}{{\text{1 + q + q}}^{\text{2}}{\text{ + q}}^{\text{3}}}\text{ = 1}$ ta có CSN: 1; 2; 4; 8

Với $\text{q = }\frac{\text{1}}{\text{2}}\Rightarrow \text{a = }\frac{\text{15}}{{\text{1 + q + q}}^{\text{2}}{\text{ + q}}^{\text{3}}}\text{ = 8}$ ta có CSN: 8; 4; 2; 1