数列と関数数列規則性、推測

学習環境

現代数学への入門微分と積分1 初等関数を中心に (青本和彦(著)、岩波書店)の第1章(数列と関数)、1.1(数列)、a(数列)の問1の解答を求めてみる。

\begin{array}{l} 1 \\ - 1 \end{array}

\begin{array}{l} \frac{1}{26 + 11} = \frac{1}{37} \\ \frac{1}{37 + 13} = \frac{1}{50} \\ \frac{1}{50 + 15} = \frac{1}{65} \\ \frac{1}{65 + 17} = \frac{1}{82} \end{array}

\begin{array}{l} 63 + 64 = 127 \\ 127 + 128 = 255 \\ 255 + 256 = 511 \\ 511 + 512 = 1023 \end{array}

\begin{array}{l} (\sqrt{6} + \sqrt{5}) + (\sqrt{7} - \sqrt{5}) = \sqrt{7} + \sqrt{6} \\ (\sqrt{7} + \sqrt{6}) + (\sqrt{8} - \sqrt{6}) = 2 \sqrt{2} + \sqrt{7} \\ (2 \sqrt{2} + \sqrt{7}) + (3 - \sqrt{7}) = 3 + 2 \sqrt{2} \\ (3 + 2 \sqrt{2}) + (\sqrt{10} - 2 \sqrt{2}) = \sqrt{10} + 3 \\ (\sqrt{10} + 3) + (\sqrt{11} - 3) = \sqrt{11} + \sqrt{10} \end{array}

\begin{array}{l} \frac{34 + 21}{34} = \frac{55}{34} \\ \frac{55 + 34}{55} = \frac{89}{55} \end{array}