Question

calcular la suma de los 10 primeros términos de la progresión geométrica cuyos tercero y cuarto son 8 y 10??

Answer 1

$Primer~Termino: a\\Termino~k: T_{k}\\Tercer~Termino: T_{3}=8\\Cuarto~Termino:T_{4}=10\\Razon: R=\frac{T_{4}}{T_{3}}\frac{10}{8}=\frac{5}{4}~~~(La~Razon~en~una~PG~se~encuentra~diviendo\\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~dos~terminos~consecutivos)\\Numero~de~Terminos:~10\\\\T_{k}=aR^{k-1}\\\\Cuando~k=3~entonces~se~T_{3}=8,~reemplacemos:\\\\T_{3}=aR^{3-1}\\8=a(\frac{5}{4})^2=a\frac{25}{16}\\\\a=8(\frac{16}{25})\\\\a=\frac{128}{25}\\\\La~Suma~de~los~10~primeros~terminos~se~calcula~de~la~siguiente\\manera:$

$\\S=a(\frac{R^{10}-1}{R-1})\\\\S=(\frac{128}{25})(\frac{(\frac{5}{4})^{10}-1}{\frac{5}{4}-1})\\\\S=(\frac{2^7}{5^2})(\frac{\frac{5^{10}-4^{10}}{4^{10}}}{\frac{1}{4}})\\\\S=(\frac{2^7}{5^2})(\frac{5^{10}-4^{10}}{4^{9}})=(\frac{2^7}{5^2})(\frac{5^{10}-4^{10}}{2^{18}})\\\\S=\frac{5^{10}-4^{10}}{(5^2)(2^{11})}\\\\S=\frac{9765625-1048576}{(25)(2048)}\\\\S=\frac{8717049}{51200}$