Question

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

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，進而解出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$\left\{\begin{array}{l}{{a}_{1}=96}\\{q=\frac{1}{2}}\end{array}\right.$．
∴方程組的解為：$\left\{\begin{array}{l}{{a}_{1}=-6}\\{q=2}\end{array}\right.$，或$\left\{\begin{array}{l}{{a}_{1}=96}\\{q=\frac{1}{2}}\end{array}\right.$．

點評 本題考查了指數(shù)冪的運算性質(zhì)、方程組的解法，考查了計算能力，屬于中檔題．