Question

Пусть
{ }n b — убывающая геометрическая прогрессия с положительными членами. Найдите ее
знаменатель, если
b4-b1=-7
b3-b2=-2

Answer 1

$\displaystyle\bf\\\left \{ {b_{4} - b_{1} =-7} \atop {b_{3}- b_{2} =-2}} \right.\\\\\\\left \{ {{b_{1} \cdot q^{3}-b_{1} =-7 } \atop {b_{1}\cdot q^{2}-b_{1} \cdot q=-2 }} \right. \\\\\\\left \{ {{b_{1} \cdot(q^{3} -1)=-7} \atop {b_{1} q\cdot(q-1)=-2}} \right. \\\\\\:\left \{ {{b_{1} \cdot(q -1)\cdot (q^{2} +q+1)=-7} \atop {b_{1} q\cdot(q-1)=-2}} \right. \\-----------------\\\frac{q^{2} +q+1}{q} =\frac{7}{2} \\\\\\2q^{2} +2q+2=7q\\\\2q^{2}-5q +2=0$

$\displaystyle\bf\\D=(-5)^{2} -4\cdot 2\cdot 2=25-16=9=3^{2} \\\\\\q_{1} =\frac{5-3}{4} =\frac{1}{2} \\\\\\q_{2} =\frac{5+3}{4} =2-neyd\\\\\\Otvet \ : \ q=\frac{1}{2}$