등비수열 곱의 합

등비수열 $\left\{ a _{n} \right\}$의 첫째항을 $a$, 공비를 $r$이라 하면 $\displaystyle a = \sqrt{2}$이므로 $a _{2} = 2$에서 $\displaystyle a _{2} = \sqrt{2} r = 2$, $\displaystyle r = \sqrt{2}$ $\therefore$ $a _{1} a _{4} + a _{2} a _{5} + a _{3} a _{6} + a _{4} a _{7}$ $= a \times ar ^{3} + ar \times ar ^{4} + ar ^{2} \times ar ^{5} + ar ^{3} \times ar ^{6}$ $= a ^{2} r ^{3} + a ^{2} r ^{5} + a ^{2} r ^{7} + a ^{2} r ^{9}$ $= a ^{2} r ^{3} \left( 1 + r ^{2} + r ^{4} + r ^{6} \right)$ $\displaystyle = 2 \times 2 \sqrt{2} \left( 1 + 2 + 4 + 8 \right)$ $\displaystyle = 60 \sqrt{2}$