Question

Obtenga los seis primeros términos de las progresiones aritméticas si: a. t4 = - 25 y t7 = 50 b. t8 = 24 y t12 = 80

Answer 1

$Primer_<span>Termino:a\\Termino~n:T_{n}\\Razon:R\\Numero~de~Termino:n\\\\T_{n}=a+(n-1)R\\\\\\a)~T_{4}=-25~y~T_{7}=50\\\\T_{4}=a+3R~~~~~~~~T_{7}=a+6R~~~~~Con~esto~tenemos~el~sistema:\\\\ \left \{ {{a+3R=-25} \atop {a+6R=~50 }} \right. ~~~~que~equivale~a~~~~\left \{ {{-a-3R=25} \atop {~~a+6R=50 }} \right.\\\\Sumemos~ambas~ecuaciones:\\\\\frac{\left \{ {{-a-3R=25} \atop {~~a+6R=50 }} \right.}{~~~~~~~3R=75}\\\\R=25,~~Luego~Reemplazamos~en~la~primera~ecuacion:\\\\a+3R=-25\\a+3(25)=-25\\a+75=-25\\a=-25-75\\a=-100$
$\\Luego:T_{n}=-100+25(n-1)=25n-25-100=25n-125\\\\-100;~-75;~-50;~-25;~0;~25$

$\\\\b)~T_{8}=24~y~T_{12}=80\\\\T_{8}=a+7R~~~~~~~~T_{12}=a+11R~~~~~Con~esto~tenemos~el~sistema:\\\\ \left \{ {{a+7R=24} \atop {a+11R=80 }} \right. ~~~~que~equivale~a~~~~\left \{ {{-a-7R=-24} \atop {a+11R=80 }} \right.\\\\Sumemos~ambas~ecuaciones:\\\\\frac{\left \{ {{-a-7R=-24} \atop {a+11R=80 }} \right.}{~~~~~~~4R=56}\\\\R=14,~~Luego~Reemplazamos~en~la~primera~ecuacion:\\\\a+7R=24\\a+7(14)=24\\a+98=24\\a=24-98\\a=-74$

$\\Luego:T_{n}=-74+14(n-1)=14n-14-74=14n-88\\\\-74;~-60;~-46;~-32;~-18;~-4$