계차수열 등비와 급수

$\displaystyle a _{n} = a _{1} + \sum\limits _{k = 1} ^{n - 1} ( a _{2} - a _{1} ) ( \frac{1}{2} ) ^{k - 1}$ $\displaystyle = - a _{1} + 2 a _{2} + ( a _{1} - a _{2} ) ( \frac{1}{2} ) ^{n - 2}$ $\displaystyle \sum\limits _{n = 1} ^{\infty} a _{n} = 20 \therefore \lim\limits _{n \rightarrow \infty} {a _{n}} = 0$ $\displaystyle \lim\limits _{n \rightarrow \infty} {\left\{ - a _{1} + 2 a _{2} + ( a _{1} - a _{2} ) \right( \frac{1}{2} ) ^{n - 2} \} = 0}$ $\displaystyle \therefore a _{2} = \frac{1}{2} a _{1}$ $\displaystyle a _{n} = ( a _{1} - a _{2} ) ( \frac{1}{2} ) ^{n - 2} = a _{1} ( \frac{1}{2} ) ^{n - 1}$ $\displaystyle \therefore \sum\limits _{n = 1} ^{\infty} a _{n} = 20 = \sum\limits _{n = 1} ^{\infty} a _{1} ( \frac{1}{2} ) ^{n - 1}$ $\displaystyle 20 = \frac{a _{1}}{1 - \frac{1}{2}}$ $\therefore a _{1} = 10$