Question

se considera progresia aritmetica (an)n>sau ega cu1 in care a9 = 6 si S9 =19 .Sa se calculeze a1 si r .Multumesc.

Answer 1

$\displaystyle a_9=6,~S_9=19,~a_1=?,~r=? \\ \\ a_9=6 \Rightarrow a_{9-1}+r=6 \Rightarrow a_8+r=6 \Rightarrow a_1+8r=6 \Rightarrow a_1=6-8r \\ \\ S_9=19 \Rightarrow \frac{2(6-8r)+(9-1) \cdot r}{2} \cdot 9=19 \Rightarrow \frac{12-16r+8r}{2} \cdot 9=19 \Rightarrow \\ \\ \Rightarrow \frac{12-8r}{2} \cdot 9=19 \Rightarrow \frac{108-72r}{2} =19 \Rightarrow 108-72r=19 \cdot 2 \Rightarrow$

$\displaystyle \Rightarrow 108-72r=38\Rightarrow -72r=38-108 \Rightarrow -72r=-70 \Rightarrow \\ \\ \Rightarrow r= \frac{70}{72} \Rightarrow r= \frac{35}{36} \\ \\ a_1=6-8r \Rightarrow a_1=6-8 \cdot \frac{35}{36} \Rightarrow a_1=6- \frac{280}{36} \Rightarrow \\ \\ \Rightarrow a_1= \frac{216}{36}- \frac{280}{36} \Rightarrow a_1=- \frac{64}{36} \Rightarrow a_1=- \frac{16}{9}$