Аритметична прогресия - системи

[alekseft]

[tex]a_{4}^{2} + a_{12}^{2} = 1170[/tex]
[tex]a_{7} + a_{15} = 60[/tex]

[tex]a_{n}=?[/tex]
d = ?

Това е система, ще съм ви благодарен, ако ми помогнете

[Добромир Глухаров]

[tex]a_n=a_1+(n-1)d;\ a_4=a_1+3d;\ a_7=a_1+6d;\ a_{12}=a_1+11d;\ a_{15}=a_1+14d[/tex]

[tex]\begin{array}{|l}(a_1+3d)^2+(a_1+11d)^2=1170\\a_1+6d+a_1+14d=60\end{array}\Leftrightarrow\begin{array}{|l}a_1=30-10d\\(30-7d)^2+(30+d)^2=1170\end{array}\Leftrightarrow\\\Leftrightarrow\begin{array}{|l}a_1=30-10d\\900-2.30.7d+49d^2+900+60d+d^2=1170\end{array}[/tex]

[tex]50d^2-360d+630=0[/tex]

[tex]5d^2-36d+63=0[/tex]

[tex]D_1=18^2-5.63=324-315=9[/tex]

[tex]d_{1,2}=\frac{18\pm\sqrt{9}}{5}[/tex]

[tex]d_1=\frac{21}{5};\ a_{1_1}=30-10\cdot\frac{21}{5}=-12\Rightarrow a_{n_1}=-12+\frac{21}{5}(n-1)=\frac{21}{5}n-\frac{81}{5}[/tex]

[tex]d_2=3;\ a_{1_2}=30-10.3=0\Rightarrow a_{n_2}=3(n-1)=3n-3[/tex]