자연계열 1-2번

[1.2] 주어진 식에서$\displaystyle \frac{11a_{10}}{10a_9}=\sum_{k=1}^9\frac{(k+2)a_{k+1}}{(k+1)a_k}-\sum_{k=1}^8\frac{(k+2)a_{k+1}}{(k+1)a_k}=(9^2+5\times9)-(8^2+5\times8)=22$이고$\displaystyle \frac{10a_9}{9a_8}=\sum_{k=1}^8\frac{(k+2)a_{k+1}}{(k+1)a_k}-\sum_{k=1}^7\frac{(k+2)a_{k+1}}{(k+1)a_k}=(8^2+5\times8)-(7^2+5\times7)=20$이므로$\displaystyle a_{10}=20a_9,\quad a_9=18a_8$이다. 따라서 $\displaystyle \frac{a_{10}}{a_8}=360$이다.