교대합 수열

수열 $\left\{ a _{n} \right\}$이 첫째항이 $20$이고 공차가 $- 3$인 등차수열이므로 $b _{2k}$$= ( a _{1} - a _{2} ) + ( a _{3} - a _{4} ) + \cdots + ( a _{2k-1} - a _{2k} )$ $= 3 + 3 + \cdots + 3$ $= 3 k$ $b _{2k-1}$$= a _{1} + ( - a _{2} + a _{3} ) + ( - a _{4} + a _{5} ) + \cdots + ( - a _{2k-2} + a _{2k-1} )$ $= 20 + ( - 3 ) + ( - 3 ) + \cdots + ( - 3 )$ $= 23 - 3 k$ ∴ $\displaystyle \sum\limits _{k=1} ^{20} b _{k}$$\displaystyle = \sum\limits _{k=1} ^{10} b _{2k-1} + \sum\limits _{k=1} ^{10} b _{2k}$ $\displaystyle = \sum\limits _{k=1} ^{10} ( b _{2k-1} + b _{2k} )$ $\displaystyle = \sum\limits _{k=1} ^{10} 23$ $= 230$