적분 수열 합

$a _{n} - a _{n-1}$$\displaystyle = \int _{0} ^{n} f ( x ) dx - \int _{0} ^{n-1} f ( x ) dx$$\displaystyle = \int _{n-1} ^{n} f ( x ) dx$ 이므로 $f ( x ) = g ( x )$일 때, $a _{1} = b _{1}$이고 $a _{n} - a _{n-1} = b _{n}$이 성립한다. ∴ $a _{10}$$\displaystyle = a _{1} + \sum\limits _{k=1} ^{9} ( a _{k+1} - a _{k} )$ $\displaystyle = b _{1} + \sum\limits _{k=1} ^{9} b _{k+1}$$\displaystyle = \sum\limits _{k=1} ^{10} b _{k}$$\displaystyle = \sum\limits _{k=1} ^{10} ( 2 k + 3 )$ (∵ $b _{n} = 2 n + 3$) $\displaystyle = \frac{10 ( 5 + 23 )}{2}$$= 140$