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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Second Half of the Chessboard ： ウィキペディア英語版 Wheat and chessboard problem The wheat and chessboard problem (sometimes expressed in terms of rice instead of wheat) is a mathematical problem in the form of a word problem: The problem may be solved using simple addition. With 64 squares on a chessboard, if the number of grains doubles on successive squares, then the sum of grains on all 64 squares is: 1 + 2 + 4 + 8... and so forth for the 64 squares. The total number of grains equals 18,446,744,073,709,551,615, much higher than what most intuitively expect. The exercise of working through this problem may be used to explain and demonstrate exponents and the quick growth of exponential and geometric sequences. It can also be used to illustrate sigma notation. When expressed as exponents, the geometric series is: 20 + 21 + 22  + 23... and so forth up to 263. The base of each exponentiation, "2", expresses the doubling at each square, while the exponents represent the position of each square (0 for the first square, 1 for the second, etc.). == Solutions == The simple, brute-force solution is to just manually double and add each step of the series: : $T\_\; =\; 1\; +\; 2\; +\; 4\; +\; \backslash cdots\; +\; 9,223,372,036,854,775,808\; =\; 18,446,744,073,709,551,615$ ::where $T\_$ is the total number of grains. The series may be expressed using exponents: : $T\_\; =\; 2^\; +\; 2^\; +\; 2^\; +\; \backslash cdots\; +\; 2^$ and, represented with capital-sigma notation as: :$\backslash sum\_^\; 2^i.\backslash ,$ It can also be solved much more easily using: : $T\_\; =\; 2^-\; 1.\; \backslash ,$ A proof of which is: : $s\; =\; 2^\; +\; 2^\; +\; 2^\; +\; \backslash cdots\; +\; 2^.$ Multiply each side by 2: : $2s\; =\; 2^\; +\; 2^\; +\; 2^\; +\; \backslash cdots\; +\; 2^\; +\; 2^.$ Subtract original series from each side: : $2s\; -\; s\; =\; -\; 2^\; +\; 2^$ : $\backslash therefore\; s\; =\; 2^-\; 1.\; \backslash ,$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Wheat and chessboard problem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.