등비수열 조건

수열 $\left\{ a _{n} \right\}$이 등비수열이므로 $a _{4} = a _{1} r ^{3}$ $a _{7} = a _{4} r ^{3}$ $a _{10} = a _{7} r ^{3}$ $\therefore$ $a _{4} + a _{7} + a _{10}$$= a _{1} r ^{3} + a _{4} r ^{3} + a _{7} r ^{3}$ $= r ^{3} \left( a _{1} + a _{4} + a _{7} \right)$ $2 \left( a _{1} + a _{4} + a _{7} \right) = a _{4} + a _{7} + a _{10}$이므로 $r ^{3} = 2$ $r ^{3} = 2$이므로 $a _{4} = a _{1} r ^{3} = 2 a _{1}$ $a _{7} = a _{4} r ^{3} = 4 a _{1}$ $a _{10} = a _{7} r ^{3} = 8 a _{1}$ $a _{4} + a _{7} + a _{10} = 6$에서 $2 a _{1} + 4 a _{1} + 8 a _{1} = 6$, $14 a _{1} = 6$ $\therefore$ $\displaystyle a _{1} = \frac{6}{14} = \frac{3}{7}$ $a _{10} = a _{1} r ^{9} = a _{1} \times \left( r ^{3} \right) ^{3}$이므로 $\displaystyle a _{10} = \frac{3}{7} \times 2 ^{3} = \frac{24}{7}$