Question

3．解方程组：$\left\{\begin{array}{l}{{a}_{1}-{a}_{1}{q}^{4}=90}\\{{a}_{1}q-{a}_{1}{q}^{3}=36}\end{array}\right.$．

Answer 1

分析 $\left\{\begin{array}{l}{{a}_{1}-{a}_{1}{q}^{4}=90}&{①}\\{{a}_{1}q-{a}_{1}{q}^{3}=36}&{②}\end{array}\right.$，②÷①可得：$\frac{q}{1+{q}^{2}}$=$\frac{2}{5}$，解得q，进而解出a₁．

解答 解：$\left\{\begin{array}{l}{{a}_{1}-{a}_{1}{q}^{4}=90}&{①}\\{{a}_{1}q-{a}_{1}{q}^{3}=36}&{②}\end{array}\right.$，
②÷①可得：$\frac{q}{1+{q}^{2}}$=$\frac{2}{5}$，解得q=2或$\frac{1}{2}$．
∴$\left\{\begin{array}{l}{{a}_{1}=-6}\\{q=2}\end{array}\right.$，或$\left\{\begin{array}{l}{{a}_{1}=96}\\{q=\frac{1}{2}}\end{array}\right.$．
∴方程组的解为：$\left\{\begin{array}{l}{{a}_{1}=-6}\\{q=2}\end{array}\right.$，或$\left\{\begin{array}{l}{{a}_{1}=96}\\{q=\frac{1}{2}}\end{array}\right.$．

点评 本题考查了指数幂的运算性质、方程组的解法，考查了计算能力，属于中档题．