로그 수열의 합

$\displaystyle a _{n} = \sum\limits _{k = 1} ^{n} a _{k} - \sum\limits _{k = 1} ^{n - 1} a _{k} = \log \frac{( n + 1 ) ( n + 2 )}{2} - \log \frac{n ( n + 1 )}{2}$ $\displaystyle = \log \frac{n + 2}{n} ( n \geq 2 )$ 이때 $n = 1$이면 $\displaystyle a _{1} = \sum\limits _{k = 1} ^{1} a _{k} = \log 3$이므로 $\displaystyle a _{n} = \log \frac{n + 2}{n} ( n \geq 1 )$ $\displaystyle \therefore p = \sum\limits _{k = 1} ^{20} a _{2 k} = \sum\limits _{k = 1} ^{20} \log \frac{2 k + 2}{2 k} = \sum\limits _{k = 1} ^{20} \log \frac{k + 1}{k}$ $\displaystyle = \log \frac{2}{1} + \log \frac{3}{2} + \cdots + \log \frac{21}{20}$ $\displaystyle = \log \left( \frac{2}{1} \times \frac{3}{2} \times \cdots \times \frac{21}{20} \right)$ $= \log 21$ $\therefore 10 ^{p} = 10 ^{\log 21} = 21$