ᱯᱷᱤᱵᱚᱱᱟᱪᱤ ᱨᱩᱞᱮᱴ ᱥᱴᱨᱮᱴᱮᱡᱤ
ᱯᱷᱤᱵᱚᱱᱟᱪᱤ ᱨᱩᱞᱮᱴ ᱥᱴᱨᱮᱴᱮᱡᱤ ᱨᱮ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱫᱚ 1, 1, 2, 3, 5, 8, 13… ᱞᱮᱠᱟᱛᱮ ᱪᱟᱞᱟᱜ-ᱟ᱾ ᱦᱟᱹᱱᱤ ᱛᱟᱭᱚᱢ ᱢᱤᱫ ᱫᱷᱟᱯ ᱢᱟᱲᱟᱝ, ᱡᱤᱛ ᱛᱟᱭᱚᱢ ᱵᱟᱨ ᱫᱷᱟᱯ ᱛᱟᱭᱚᱢ᱾
ᱱᱚᱶᱟ ᱫᱚ ᱪᱟᱠᱟ ᱵᱟᱭ ᱦᱟᱨᱟᱣᱟ; ᱡᱟᱦᱟᱸᱱ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱯᱨᱚᱡᱚᱱ ᱦᱚᱸ ᱵᱟᱭ ᱦᱟᱨᱟᱣᱟ᱾ ᱱᱚᱶᱟ ᱫᱚ ᱢᱟᱨᱴᱤᱝᱜᱮᱞ ᱠᱷᱚᱱ ᱞᱟᱦᱟᱭ ᱜᱮᱭᱟ, ᱚᱱᱟᱛᱮ ᱥᱤᱢᱟᱹ ᱥᱮᱴᱮᱨ ᱞᱟᱦᱟ ᱡᱮᱞᱮᱧ ᱦᱟᱹᱱᱤ ᱞᱟᱜᱟᱛᱟᱨ ᱥᱚᱦᱚᱫ ᱫᱟᱲᱮᱭᱟᱜ-ᱟ, ᱟᱨ ᱞᱟᱦᱟᱭ ᱛᱮ ᱨᱩᱣᱟᱹᱲᱚᱜ-ᱟ᱾ ᱟᱸᱫᱟᱡᱽ ᱦᱟᱹᱱᱤ ᱫᱚ ᱡᱚᱛᱚ ᱵᱟᱡᱤ ᱦᱤᱥᱟᱹᱵᱴᱟᱠᱟᱡᱚᱛᱚ ᱵᱟᱡᱤ ᱦᱤᱥᱟᱹᱵᱢᱤᱫ ᱥᱮᱥᱚᱱ ᱟᱨᱵᱟᱝ ᱚᱠᱛᱚ ᱨᱮ ᱡᱚᱛᱚ ᱵᱟᱡᱤ ᱴᱟᱠᱟ, ᱫᱚᱦᱲᱟ ᱞᱟᱜᱟᱣ ᱡᱤᱛ ᱴᱟᱠᱟ ᱥᱮᱞᱮᱫ᱾ ᱮᱴᱟᱜ ᱧᱩᱛᱩᱢ: ᱦᱮᱱᱰᱮᱞ, ᱮᱠᱥᱚᱱ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱨᱮᱭᱟᱜ ᱚᱱᱟ ᱜᱮ 2.70% ᱠᱟᱱᱟ᱾
ᱥᱤᱨᱤᱡᱽ ᱪᱮᱫ ᱞᱮᱠᱟ ᱵᱮᱱᱟᱣᱚᱜ-ᱟ
ᱢᱤᱫ ᱢᱤᱫ ᱮᱞ ᱫᱚ ᱟᱡ ᱞᱟᱦᱟ ᱨᱮᱭᱟᱜ ᱵᱟᱨ ᱮᱞ ᱨᱮᱭᱟᱜ ᱡᱚᱛᱚ ᱠᱟᱱᱟ᱾
ᱚᱱᱛᱮ 8 ᱟᱨ 13 ᱛᱟᱭᱚᱢ 21 ᱦᱮᱡᱚᱜ-ᱟ᱾
ᱢᱤᱫ ᱢᱤᱫ ᱮᱞ ᱪᱟᱹᱨᱠᱚᱬ ᱞᱮᱠᱟ ᱚᱸᱠᱟᱣ ᱯᱮ, ᱟᱨ ᱪᱟᱹᱨᱠᱚᱬ ᱠᱚ ᱢᱤᱫ ᱥᱟᱶ ᱥᱮᱴᱟᱣᱚᱜ-ᱟ: ᱡᱚᱛᱚ ᱱᱟᱶᱟ ᱪᱟᱹᱨᱠᱚᱬ ᱫᱚ ᱟᱡ ᱞᱟᱦᱟ ᱨᱮᱭᱟᱜ ᱵᱟᱨ ᱪᱟᱹᱨᱠᱚᱬ ᱫᱷᱟᱨᱮ ᱫᱷᱟᱨᱮ ᱫᱚᱦᱚ ᱞᱮᱠᱟ ᱡᱮᱞᱮᱧ ᱠᱟᱱᱟ᱾ ᱢᱤᱫ ᱢᱤᱫ ᱪᱟᱹᱨᱠᱚᱬ ᱨᱮᱭᱟᱜ ᱯᱟᱣ ᱜᱩᱞᱟᱹᱭ ᱫᱚ ᱢᱤᱫ ᱥᱟᱶ ᱡᱚᱲᱟᱣ ᱠᱟᱛᱮ ᱢᱤᱫ ᱜᱷᱩᱨᱤ ᱦᱚᱨ ᱵᱮᱱᱟᱣᱚᱜ-ᱟ᱾
ᱥᱤᱨᱤᱡᱽ ᱛᱮ ᱥᱮᱱ ᱯᱮ
ᱦᱟᱹᱱᱤ ᱫᱚ ᱢᱤᱫ ᱫᱷᱟᱯ ᱡᱚᱡᱚᱢ ᱛᱮ ᱥᱟᱦᱮᱫ ᱮᱫᱟ᱾ ᱡᱤᱛ ᱫᱚ ᱵᱟᱨ ᱫᱷᱟᱯ ᱛᱟᱭᱚᱢ ᱛᱮ ᱥᱟᱦᱮᱫ ᱮᱫᱟ᱾
ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱟᱨ ᱨᱮᱡᱟᱞᱴ ᱫᱚ ᱢᱩᱞ ᱵᱟᱡᱤ ᱨᱮᱭᱟᱜ ᱭᱩᱱᱤᱴ ᱨᱮ ᱢᱮᱱᱟᱜ-ᱟ᱾
ᱯᱷᱤᱵᱚᱱᱟᱪᱤ ᱨᱩᱞᱮᱴ ᱥᱤᱥᱴᱮᱢ ᱪᱮᱫ ᱞᱮᱠᱟ ᱠᱟᱹᱢᱤᱭ-ᱟ?
ᱯᱷᱤᱵᱚᱱᱟᱪᱤᱥᱴᱨᱮᱴᱮᱡᱤᱯᱷᱤᱵᱚᱱᱟᱪᱤ1, 1, 2, 3, 5, 8, 13… ᱥᱤᱨᱤᱡᱽ ᱯᱟᱧᱡᱟ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ: ᱦᱟᱨᱟᱣ ᱛᱟᱭᱚᱢ ᱢᱤᱫ ᱫᱷᱟᱯ ᱞᱟᱦᱟ, ᱡᱤᱛ ᱛᱟᱭᱚᱢ ᱵᱟᱨ ᱫᱷᱟᱯ ᱛᱟᱭᱚᱢ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱮᱞ ᱠᱚ ᱚᱞ ᱯᱮ: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144᱾ ᱡᱚᱛᱚ ᱮᱞ ᱫᱚ ᱞᱟᱦᱟ ᱨᱮᱭᱟᱜ ᱵᱟᱨ ᱮᱞ ᱨᱮᱭᱟᱜ ᱡᱚᱲ ᱠᱟᱱᱟ᱾ ᱯᱟᱦᱤᱞ ᱮᱞ ᱠᱷᱚᱱ ᱮᱦᱚᱵ ᱯᱮ ᱟᱨ ᱚᱛᱮᱠ ᱭᱩᱱᱤᱴᱴᱟᱠᱟᱭᱩᱱᱤᱴᱢᱤᱫ ᱥᱤᱥᱴᱮᱢ ᱟᱨᱵᱟᱝ ᱠᱷᱮᱞᱚᱸᱰ ᱵᱮᱵᱷᱟᱨ ᱵᱮᱥ ᱵᱟᱡᱤ ᱴᱟᱠᱟ᱾ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ ᱠᱚ ᱭᱩᱱᱤᱴ ᱛᱮ ᱵᱩᱡᱷᱟᱹᱣᱚᱜ-ᱟ ᱡᱮᱢᱚᱱ ᱡᱟᱦᱟᱸᱱ ᱴᱮᱵᱩᱞ ᱨᱮ ᱞᱟᱹᱜᱩ ᱦᱩᱭᱩᱜ-ᱟ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱵᱟᱡᱤ ᱞᱟᱜᱟᱣ ᱯᱮ᱾ ᱦᱟᱹᱱᱤ ᱞᱮᱠᱷᱟᱱ ᱢᱤᱫ ᱫᱷᱟᱯ ᱡᱚᱡᱚᱢ ᱥᱮᱫ ᱥᱟᱨᱮᱡ ᱯᱮ᱾ ᱡᱤᱛ ᱞᱮᱠᱷᱟᱱ ᱵᱟᱨ ᱫᱷᱟᱯ ᱞᱮᱸᱜᱟ ᱥᱮᱫ ᱥᱟᱨᱮᱡ ᱯᱮ, ᱟᱨᱵᱟᱝ ᱡᱟᱭᱜᱟ ᱵᱟᱹᱱᱩᱜ ᱞᱮᱠᱷᱟᱱ ᱮᱦᱚᱵ ᱨᱮ ᱨᱩᱣᱟᱹᱲ ᱯᱮ᱾ ᱚᱱᱟ ᱜᱮ ᱡᱚᱛᱚ ᱥᱤᱥᱴᱮᱢ ᱠᱟᱱᱟ᱾ ᱴᱮᱵᱩᱞ ᱫᱚ ᱡᱚᱛᱚ ᱫᱷᱟᱯ ᱨᱮᱭᱟᱜ ᱠᱷᱚᱨᱚᱪ ᱩᱫᱩᱜ ᱮᱫᱟ: ᱴᱮᱵᱩᱞ ᱥᱤᱢᱟᱹᱴᱮᱵᱩᱞᱴᱮᱵᱩᱞ ᱥᱤᱢᱟᱹᱦᱩᱠᱩᱢ ᱢᱮᱱᱟᱜ ᱠᱚᱢ ᱥᱮ ᱠᱚᱢ ᱟᱨ ᱵᱮᱥᱤ ᱥᱮ ᱵᱮᱥᱤ ᱵᱟᱡᱤ᱾ ᱵᱮᱥᱤ ᱥᱮ ᱵᱮᱥᱤ ᱥᱤᱢᱟᱹ ᱜᱮ ᱰᱚᱵᱚᱞ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ ᱵᱚᱸᱫᱟ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱥᱮ ᱟᱯᱮᱭᱟᱜ ᱠᱷᱮᱞ ᱯᱩᱸᱡᱤᱴᱟᱠᱟᱠᱷᱮᱞ ᱯᱩᱸᱡᱤᱠᱷᱮᱞ ᱞᱟᱹᱜᱤᱫ ᱟᱞᱟᱜ ᱫᱚᱦᱚ ᱟᱠᱟᱱ ᱡᱚᱛᱚ ᱴᱟᱠᱟ᱾ ᱠᱷᱮᱞ ᱯᱩᱸᱡᱤ ᱥᱟᱢᱵᱷᱟᱞ ᱫᱚ ᱢᱮᱱᱮᱛ ᱢᱤᱫ ᱵᱟᱡᱤ ᱟᱨ ᱢᱤᱫ ᱥᱮᱥᱚᱱ ᱨᱮ ᱚᱱᱟ ᱠᱷᱚᱱ ᱛᱤᱱᱟᱹᱜ ᱡᱚᱠᱷᱤᱢ ᱨᱮ ᱫᱚᱦᱚᱭ ᱚᱱᱟ ᱴᱷᱟᱹᱣᱠᱟᱹ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱦᱚᱛᱮᱛᱮ ᱛᱷᱤᱨᱚᱜ ᱦᱟᱹᱱᱤ ᱞᱟᱜᱟᱛᱟᱨ ᱫᱚ ᱱᱤᱛ ᱦᱟᱹᱵᱤᱡ ᱡᱚᱛᱚ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱨᱮᱭᱟᱜ ᱡᱚᱲ ᱦᱟᱹᱱᱤ ᱮᱫᱟ᱾
ᱢᱤᱫ ᱥᱮᱥᱚᱱ ᱫᱚ ᱪᱮᱫ ᱞᱮᱠᱟ ᱧᱮᱞᱚᱜ-ᱟ?
$10 ᱵᱮᱥ ᱟᱨ ᱤᱨᱟᱹᱞ ᱜᱷᱩᱨᱟᱹᱣ ᱦᱟᱛᱟᱣ ᱯᱮ: ᱦ ᱦ ᱡ ᱦ ᱦ ᱦ ᱡ ᱡ (ᱦ = ᱦᱟᱹᱱᱤ, ᱡ = ᱡᱤᱛ)᱾ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱫᱚ 10, 10, 20 (ᱡᱤᱛ, ᱵᱟᱨ ᱫᱷᱟᱯ ᱛᱟᱭᱚᱢ, ᱫᱷᱟᱯ 1 ᱛᱮ), 10, 10, 20, 30 (ᱡᱤᱛ, ᱵᱟᱨ ᱫᱷᱟᱯ ᱛᱟᱭᱚᱢ, ᱫᱷᱟᱯ 2 ᱛᱮ), 10 (ᱡᱤᱛ, ᱮᱦᱚᱵ ᱛᱮ ᱨᱩᱣᱟᱹᱲ) ᱞᱮᱠᱟᱛᱮ ᱪᱟᱞᱟᱜ-ᱟ᱾ ᱦᱟᱹᱱᱤ: 10 + 10 + 10 + 10 + 20 = 60᱾ ᱡᱤᱛ: 20 + 30 + 10 = 60᱾ ᱢᱚᱬᱮ ᱦᱟᱹᱱᱤ, ᱯᱮ ᱡᱤᱛ, ᱟᱨ ᱟᱯᱮ ᱵᱟᱨᱟᱵᱟᱨ ᱨᱮ ᱢᱮᱱᱟᱢ ᱯᱮ᱾ ᱚᱱᱟ ᱜᱮ ᱥᱤᱥᱴᱮᱢ ᱨᱮᱭᱟᱜ ᱞᱩᱵᱷᱟᱹᱣ ᱠᱟᱱᱟ᱾ ᱠᱷᱚᱨᱚᱪ ᱫᱚ ᱚᱱᱟ ᱜᱟᱨ ᱨᱮ ᱜᱮ ᱢᱮᱱᱟᱜ-ᱟ: ᱚᱸᱰᱮ ᱥᱮᱴᱮᱨ ᱞᱟᱹᱜᱤᱫ $120 ᱵᱟᱡᱤ ᱞᱟᱜᱟᱣ ᱞᱟᱹᱠᱛᱤ ᱮᱱᱟ, ᱟᱨ ᱵᱟᱡᱤ ᱞᱟᱜᱟᱣ ᱡᱚᱛᱚ ᱰᱚᱞᱟᱨ ᱨᱮ ᱠᱮᱥᱤᱱᱳ ᱵᱟᱲᱛᱤᱜᱚᱱᱤᱛᱠᱮᱥᱤᱱᱳ ᱵᱟᱲᱛᱤᱡᱚᱛᱚ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱠᱷᱚᱱ ᱠᱮᱥᱤᱱᱳ ᱟᱣᱥᱚᱛ ᱛᱮ ᱫᱚᱦᱚᱭ ᱦᱟᱹᱴᱤᱧ: ᱭᱩᱨᱚᱯᱤᱭᱟᱱ ᱨᱩᱞᱮᱴ ᱨᱮ 2.70%᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱢᱮᱱᱟᱜ-ᱟ᱾
ᱵᱟᱨ ᱫᱷᱟᱯ ᱪᱮᱫᱟᱜ ᱛᱟᱭᱚᱢ, ᱢᱤᱫ ᱫᱚ ᱪᱮᱫᱟᱜ ᱵᱟᱝ?
ᱪᱮᱫᱟᱜ ᱥᱮ ᱢᱤᱫ ᱡᱤᱛ ᱫᱚ ᱴᱷᱤᱠ ᱵᱟᱨ ᱦᱟᱹᱱᱤ ᱨᱮᱭᱟᱜ ᱫᱟᱢ ᱮᱢ ᱮᱫᱟ᱾ ᱡᱚᱛᱚ ᱮᱞ ᱫᱚ ᱞᱟᱦᱟ ᱨᱮᱭᱟᱜ ᱵᱟᱨ ᱮᱞ ᱨᱮᱭᱟᱜ ᱡᱚᱲ ᱠᱟᱱᱟ, ᱚᱱᱟᱛᱮ 5 ᱭᱩᱱᱤᱴ ᱨᱮ ᱡᱤᱛ ᱫᱚ ᱞᱟᱦᱟ ᱨᱮ ᱦᱟᱹᱱᱤ 2 ᱟᱨ 3 ᱨᱩᱣᱟᱹᱲ ᱮᱫᱟ᱾ ᱢᱤᱫ ᱥᱤᱲᱦᱤ ᱩᱭᱦᱟᱹᱨ ᱯᱮ: ᱡᱚᱛᱚ ᱦᱟᱹᱱᱤ ᱞᱟᱹᱜᱤᱫ ᱢᱤᱫ ᱫᱷᱟᱯ ᱪᱮᱛᱟᱱ, ᱡᱚᱛᱚ ᱡᱤᱛ ᱞᱟᱹᱜᱤᱫ ᱵᱟᱨ ᱫᱷᱟᱯ ᱞᱟᱛᱟᱨ᱾ ᱟᱯᱮ ᱡᱟᱦᱟᱸ ᱫᱷᱟᱯ ᱨᱮ ᱛᱤᱝᱜᱩ ᱢᱮᱱᱟᱢ ᱯᱮ ᱚᱱᱟ ᱫᱚ ᱟᱯᱮᱭᱟᱜ ᱨᱤᱱ ᱠᱟᱱᱟ᱾ ᱫᱷᱟᱯ 5 ᱨᱮ ᱟᱯᱮ 7 ᱭᱩᱱᱤᱴ ᱦᱟᱹᱱᱤ ᱨᱮ ᱢᱮᱱᱟᱢ ᱯᱮ, ᱫᱷᱟᱯ 8 ᱨᱮ 33 ᱦᱟᱹᱱᱤ ᱨᱮ, ᱟᱨ ᱯᱟᱦᱤᱞ ᱫᱷᱟᱯ ᱨᱮ ᱨᱩᱣᱟᱹᱲ ᱠᱟᱛᱮ ᱴᱷᱤᱠ ᱵᱟᱨᱟᱵᱟᱨ᱾ ᱞᱟᱵᱷ ᱫᱚ ᱯᱟᱦᱤᱞ ᱫᱷᱟᱯ ᱨᱮᱭᱟᱜ ᱡᱤᱛ ᱠᱷᱚᱱ ᱜᱮ ᱦᱮᱡᱚᱜ-ᱟ, ᱢᱤᱫ ᱫᱷᱟᱣ ᱨᱮ ᱢᱤᱫ ᱭᱩᱱᱤᱴ᱾
ᱦᱟᱹᱱᱤ ᱞᱮᱠᱟ ᱡᱚᱦᱚᱢ ᱡᱤᱛ ᱦᱩᱭ ᱠᱷᱟᱱ, ᱢᱤᱫ ᱞᱟᱹᱜᱤᱫ ᱵᱟᱨ ᱫᱷᱟᱯ ᱫᱚ ᱵᱮᱥ ᱥᱟᱣᱫᱟ ᱞᱮᱠᱟ ᱟᱹᱭᱩᱨᱚᱜ-ᱟ, ᱟᱨ ᱡᱟᱹᱥᱛᱤ ᱚᱠᱛᱚ ᱥᱤᱲᱦᱤ ᱫᱚ ᱞᱟᱛᱟᱨ ᱜᱮ ᱟᱹᱜᱩᱭ-ᱟ᱾ ᱯᱮᱸᱪ ᱫᱚ ᱱᱚᱶᱟ ᱠᱟᱱᱟ ᱡᱮ ᱪᱮᱛᱟᱱ ᱥᱮᱫ ᱪᱟᱲᱦᱟᱣ ᱞᱮᱠᱷᱟᱱ ᱫᱷᱟᱯ ᱠᱚ ᱩᱥᱩᱞᱚᱜ-ᱟ᱾ ᱯᱟᱦᱤᱞ ᱫᱷᱟᱯ ᱨᱮ ᱜᱮᱞ ᱡᱤᱛ ᱫᱚ 10 ᱭᱩᱱᱤᱴ ᱮᱢ ᱮᱫᱟ᱾ ᱫᱷᱟᱯ 6, 7 ᱟᱨ 8 ᱨᱮ ᱯᱮ ᱦᱟᱹᱱᱤ ᱫᱚ 8 + 13 + 21 = 42 ᱠᱷᱚᱨᱚᱪᱚᱜ-ᱟ᱾
- p10 −$500
- p25 −$500
- ᱢᱤᱰᱤᱭᱟᱱ −$500
- p75 +$250
- p90 +$920
ᱯᱷᱤᱵᱚᱱᱟᱪᱤ ᱥᱤᱥᱴᱮᱢ ᱠᱟᱹᱢᱤᱭ-ᱟ ᱥᱮ?
ᱵᱟᱝ᱾ ᱞᱟᱵᱷ ᱥᱟᱶ ᱵᱟᱦᱨᱮ ᱦᱚᱨᱟ ᱞᱮᱠᱟ ᱫᱚ ᱵᱟᱝ᱾ ᱦᱤᱥᱟᱹᱵ ᱫᱚ ᱡᱚᱛᱚ ᱥᱤᱥᱴᱮᱢ ᱞᱮᱠᱟ ᱜᱮ: ᱡᱚᱛᱚ ᱵᱟᱡᱤ ᱫᱚ ᱟᱣᱥᱚᱛ ᱛᱮ ᱟᱡᱟᱜ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱨᱮᱭᱟᱜ 1/37 ᱦᱟᱹᱱᱤ ᱮᱫᱟ, ᱚᱱᱟᱛᱮ ᱡᱚᱛᱚ ᱫᱷᱟᱨᱟ ᱫᱚ ᱡᱚᱛᱚ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱨᱮᱭᱟᱜ 1/37 ᱦᱟᱹᱱᱤ ᱮᱫᱟ᱾ ᱨᱩᱞᱮᱴ ᱨᱮ ᱯᱷᱤᱵᱚᱱᱟᱪᱤ ᱫᱷᱟᱨᱟ ᱵᱮᱵᱷᱟᱨ ᱫᱚ ᱚᱱᱟ ᱵᱟᱭ ᱵᱚᱫᱚᱞᱟ, ᱪᱮᱫᱟᱜ ᱥᱮ ᱪᱤᱯᱴᱟᱠᱟᱪᱤᱯᱛᱷᱤᱠ ᱴᱟᱠᱟ ᱨᱮᱭᱟᱜ ᱪᱤᱱᱦᱟᱹ ᱴᱚᱠᱮᱱ, ᱵᱟᱡᱤ ᱞᱟᱹᱜᱤᱫ ᱞᱮᱭᱟᱣᱴ ᱨᱮ ᱫᱚᱦᱚᱜ-ᱟ᱾ ᱮᱴᱟᱜ ᱧᱩᱛᱩᱢ: ᱪᱮᱠ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱨᱮᱭᱟᱜ ᱢᱟᱯ ᱡᱟᱦᱟᱸ ᱦᱩᱭ ᱦᱚᱸ ᱪᱟᱠᱟ ᱫᱚ ᱚᱱᱟ ᱜᱮ ᱚᱰᱥ ᱮᱢ ᱮᱫᱟ᱾ ᱯᱷᱤᱵᱚᱱᱟᱪᱤ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱫᱚ ᱪᱮᱛᱟᱱ ᱥᱮᱫ ᱥᱟᱨᱮᱡᱚᱜ-ᱟ, ᱚᱱᱟᱛᱮ ᱱᱚᱶᱟ ᱫᱚ ᱢᱤᱫ ᱞᱮᱠᱟᱱ ᱵᱟᱡᱤᱥᱴᱨᱮᱴᱮᱡᱤᱢᱤᱫ ᱞᱮᱠᱟᱱ ᱵᱟᱡᱤᱡᱚᱛᱚ ᱜᱷᱩᱨᱟᱹᱣ ᱨᱮ ᱢᱤᱫ ᱞᱮᱠᱟᱱ ᱴᱟᱠᱟ ᱵᱟᱡᱤ᱾ ᱡᱚᱛᱚ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ ᱱᱚᱶᱟ ᱥᱟᱶ ᱛᱩᱞᱟᱹᱣᱚᱜ-ᱟ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱠᱷᱚᱱ ᱢᱤᱫ ᱜᱷᱩᱨᱟᱹᱣ ᱨᱮ ᱵᱮᱥᱤ ᱵᱟᱡᱤ ᱞᱟᱜᱟᱣ ᱮᱫᱟ, ᱟᱨ ᱢᱤᱫ ᱥᱮᱥᱚᱱ ᱨᱮ ᱚᱱᱟ ᱨᱮᱭᱟᱜ ᱟᱸᱫᱟᱡᱽ ᱦᱟᱹᱱᱤᱜᱚᱱᱤᱛᱟᱸᱫᱟᱡᱽ ᱦᱟᱹᱱᱤᱟᱸᱫᱟᱡᱽ ᱜᱚᱱᱚᱝ ᱜᱩᱬ ᱡᱚᱛᱚ ᱵᱟᱡᱤ ᱴᱟᱠᱟ; ᱢᱤᱫ ᱥᱮᱥᱚᱱ ᱟᱨᱵᱟᱝ ᱚᱠᱛᱚ ᱨᱮ ᱟᱣᱥᱚᱛ ᱛᱮ ᱦᱟᱨᱟᱣᱚᱜ ᱴᱟᱠᱟ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱵᱮᱥᱤ ᱜᱮᱭᱟ᱾ ᱢᱟᱨᱴᱤᱝᱜᱮᱞᱥᱴᱨᱮᱴᱮᱡᱤᱢᱟᱨᱴᱤᱝᱜᱮᱞᱡᱚᱛᱚ ᱦᱟᱨᱟᱣ ᱛᱟᱭᱚᱢ ᱵᱟᱡᱤ ᱰᱚᱵᱚᱞ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ᱾ ᱱᱚᱶᱟ ᱪᱮᱫ ᱮ ᱠᱟᱹᱢᱤᱭᱟ ᱟᱨ ᱪᱮᱫ ᱵᱟᱭ ᱠᱟᱹᱢᱤᱭᱟ, ᱚᱱᱟ ᱥᱴᱨᱮᱴᱮᱡᱤ ᱦᱟᱹᱴᱤᱧ ᱨᱮ ᱧᱮᱞ ᱯᱮ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱫᱚ ᱟᱨᱦᱚᱸ ᱵᱮᱥᱤ ᱵᱟᱡᱤ ᱞᱟᱜᱟᱣ ᱮᱫᱟ, ᱟᱨ ᱵᱮᱥᱤ ᱦᱟᱹᱱᱤ ᱮᱫᱟ᱾
ᱱᱚᱶᱟ ᱫᱚ ᱢᱟᱨᱴᱤᱝᱜᱮᱞ ᱠᱷᱚᱱ ᱵᱮᱥᱤ ᱨᱚᱠᱷᱤᱭᱟ ᱜᱮᱭᱟ ᱥᱮ?
ᱱᱚᱶᱟ ᱫᱚ ᱞᱟᱦᱟᱭ ᱜᱮᱭᱟ, ᱟᱨ ᱚᱱᱟ ᱫᱚ ᱢᱤᱫ ᱠᱟᱛᱷᱟ ᱵᱟᱝ ᱠᱟᱱᱟ᱾ ᱯᱷᱤᱵᱚᱱᱟᱪᱤ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱫᱚ ᱡᱚᱛᱚ ᱫᱷᱟᱯ ᱨᱮ 2× ᱵᱚᱫᱚᱞ ᱨᱮ ᱡᱚᱦᱚᱢ 1.6× ᱵᱟᱲᱦᱟᱣᱚᱜ-ᱟ, ᱚᱱᱟᱛᱮ $10 ᱵᱮᱥ ᱠᱷᱚᱱ $500 ᱥᱤᱢᱟᱹ ᱫᱚ ᱛᱩᱨᱩᱭ ᱵᱚᱫᱚᱞ ᱨᱮ ᱡᱚᱦᱚᱢ ᱜᱮᱞ ᱞᱟᱜᱟᱛᱟᱨ ᱦᱟᱹᱱᱤ ᱛᱟᱭᱚᱢ ᱥᱮᱴᱮᱨᱚᱜ-ᱟ᱾ ᱚᱱᱟᱛᱮ ᱵᱩᱨᱩ ᱠᱷᱚᱱ ᱧᱩᱨ ᱫᱚ ᱵᱮᱥᱤ ᱜᱟᱱ ᱜᱟᱱ ᱦᱩᱭᱩᱜ-ᱟ: ᱞᱟᱜᱟᱛᱟᱨ ᱜᱮᱞ ᱥᱚᱢᱟᱱ ᱴᱟᱠᱟ ᱦᱟᱹᱱᱤ ᱫᱚ (19/37)¹⁰ ≈ 0.13% ᱠᱟᱱᱟ, ᱛᱩᱨᱩᱭ ᱞᱟᱹᱜᱤᱫ (19/37)⁶ ≈ 1.8% ᱛᱩᱞᱟᱹ᱾ ᱢᱮᱱᱠᱷᱟᱱ ᱵᱩᱨᱩ ᱦᱚᱸ ᱵᱮᱥᱤ ᱩᱥᱩᱞ ᱜᱮᱭᱟ, ᱪᱮᱫᱟᱜ ᱥᱮ ᱚᱸᱰᱮ ᱥᱮᱴᱮᱨ ᱦᱚᱨᱟ ᱨᱮ ᱵᱮᱥᱤ ᱵᱟᱡᱤ ᱞᱟᱜᱟᱣ ᱦᱩᱭ ᱟᱠᱟᱱᱟ, ᱟᱨ ᱢᱤᱫ ᱡᱤᱛ ᱫᱚ ᱚᱱᱟ ᱵᱟᱭ ᱢᱮᱴᱟᱣᱟ᱾
ᱮᱞ ᱠᱚ ᱚᱠᱟ ᱠᱷᱚᱱ ᱦᱮᱡ ᱟᱠᱟᱱᱟ?
ᱠᱩᱞᱟᱹᱭ ᱵᱟᱵᱚᱫ ᱢᱤᱫ ᱠᱩᱠᱞᱤ ᱠᱷᱚᱱ᱾ ᱯᱤᱥᱟ ᱨᱮᱭᱟᱜ ᱞᱤᱭᱚᱱᱟᱨᱰᱳ (Leonardo of Pisa), ᱤᱧᱟᱹ ᱛᱟᱭᱚᱢ ᱯᱷᱤᱵᱚᱱᱟᱪᱤ ᱧᱩᱛᱩᱢ ᱛᱮ ᱵᱟᱰᱟᱭᱚᱜ, ᱚᱱᱟ 1202 ᱥᱮᱨᱢᱟ ᱨᱮᱭᱟᱜ ᱟᱡᱟᱜ Liber Abaci ᱨᱮ ᱮ ᱚᱞ ᱞᱮᱫᱟ: ᱢᱤᱫ ᱡᱚᱲᱟ ᱥᱟᱶ ᱮᱦᱚᱵ ᱯᱮ, ᱡᱚᱛᱚ ᱡᱚᱲᱟ ᱫᱚ ᱟᱡᱟᱜ ᱫᱚᱥᱟᱨ ᱪᱟᱸᱫᱚ ᱠᱷᱚᱱ ᱡᱚᱛᱚ ᱪᱟᱸᱫᱚ ᱢᱤᱫ ᱱᱟᱣᱟ ᱡᱚᱲᱟ ᱮᱢ ᱟᱭ, ᱟᱨ ᱡᱚᱲᱟ ᱠᱚ ᱞᱮᱠᱷᱟ ᱯᱮ᱾ ᱛᱮᱞᱟ ᱫᱚ 1, 2, 3, 5, 8, 13 ᱞᱮᱠᱟᱛᱮ ᱪᱟᱞᱟᱜ-ᱟ᱾ ᱩᱱᱤ ᱫᱚ ᱱᱤᱭᱚᱢ ᱧᱟᱢ ᱤᱫᱤ ᱯᱟᱦᱤᱞ ᱦᱚᱲ ᱫᱚ ᱵᱟᱝ ᱛᱟᱦᱮᱸ ᱠᱟᱱᱟᱭ᱾ ᱥᱮᱨᱮᱧ ᱨᱮᱭᱟᱜ ᱞᱟᱭ ᱵᱟᱵᱚᱫ ᱥᱮᱱᱟᱢ ᱤᱱᱰᱤᱭᱟᱱ ᱯᱚᱸᱰᱤᱛ ᱠᱚ ᱞᱟᱦᱟ ᱜᱮ ᱚᱞ ᱟᱠᱟᱫᱟ ᱠᱚ: ᱥᱤᱝ (Singh) (1985) ᱫᱚ 600 ᱟᱨ 800 ᱛᱟᱞᱟ ᱨᱮ ᱵᱤᱨᱟᱦᱟᱝᱠᱟ (Virahanka) ᱦᱟᱹᱵᱤᱡ, ᱟᱨ ᱛᱟᱭᱚᱢ ᱜᱚᱯᱟᱞ (Gopala) ᱟᱨ ᱦᱮᱢᱚᱪᱚᱸᱫᱽᱨᱚ (Hemachandra) ᱦᱟᱹᱵᱤᱡ ᱧᱟᱢ ᱟᱠᱟᱫᱟᱭ᱾
ᱚᱱᱟ ᱠᱚ ᱛᱮ ᱡᱟᱦᱟᱸᱭ ᱦᱚᱸ ᱢᱤᱫ ᱪᱟᱠᱟ ᱵᱟᱵᱚᱫ ᱵᱟᱝ ᱩᱭᱦᱟᱹᱨ ᱞᱮᱫᱟ᱾ ᱱᱚᱸᱰᱮ ᱵᱮᱵᱷᱟᱨ ᱯᱷᱮᱰᱟᱛ ᱠᱚ ᱫᱚ ᱵᱟᱝ ᱢᱮᱛᱟᱜ-ᱟ ᱡᱮ ᱚᱠᱟᱭ ᱯᱟᱦᱤᱞ ᱱᱚᱶᱟ ᱫᱷᱟᱨᱟ ᱛᱮ ᱴᱟᱠᱟ ᱵᱟᱡᱤ ᱞᱟᱜᱟᱣ ᱞᱮᱫᱟ᱾ ᱡᱟᱦᱟᱸ ᱩᱫᱩᱜ ᱮᱫᱟ ᱚᱱᱟ ᱫᱚ ᱱᱚᱶᱟ ᱠᱟᱱᱟ ᱡᱮ ᱚᱱᱟ ᱨᱮᱭᱟᱜ ᱦᱤᱥᱟᱹᱵ ᱟᱨ ᱨᱩᱞᱮᱴ ᱨᱮᱭᱟᱜ ᱦᱤᱥᱟᱹᱵ ᱫᱚ ᱛᱤᱥ ᱦᱚᱸ ᱵᱟᱝ ᱢᱤᱞᱟᱹᱣᱚᱜ-ᱟ: ᱢᱤᱫ ᱫᱚ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱪᱮᱫ ᱞᱮᱠᱟ ᱵᱟᱲᱦᱟᱣᱚᱜ-ᱟ ᱚᱱᱟ ᱢᱮᱛᱟᱜ-ᱟ, ᱮᱴᱟᱜ ᱫᱚ ᱡᱚᱛᱚ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱪᱮᱫ ᱨᱩᱣᱟᱹᱲ ᱮᱫᱟ ᱚᱱᱟ ᱢᱮᱛᱟᱜ-ᱟ᱾
ᱢᱤᱫ ᱦᱟᱹᱱᱤ ᱞᱟᱜᱟᱛᱟᱨ ᱛᱟᱭᱚᱢ ᱛᱤᱢᱤᱱ ᱦᱟᱹᱱᱤᱚᱜ-ᱟ?
ᱵᱟᱨ ᱜᱚᱴᱟᱝ ᱥᱟᱹᱨᱤ ᱥᱚᱢᱤᱠᱟᱨᱚᱬ ᱪᱮᱛᱟᱱ ᱨᱮᱭᱟᱜ ᱴᱮᱵᱩᱞ ᱪᱮᱠᱟᱣ ᱟᱞᱜᱟ ᱮᱫᱟ᱾ ᱞᱟᱜᱟᱛᱟᱨ n ᱦᱟᱹᱱᱤ ᱛᱟᱭᱚᱢ ᱡᱚᱛᱚ ᱦᱟᱹᱱᱤ ᱫᱚ (n + 2) ᱮᱞ ᱯᱷᱤᱵᱚᱱᱟᱪᱤ ᱮᱞ ᱠᱷᱚᱱ 1 ᱠᱚᱢ ᱠᱟᱱᱟ, ᱡᱟᱦᱟᱸ ᱫᱚ 1, 2, 4, 7, 12, 20 ᱟᱨ ᱮᱢᱟᱱ ᱮᱢ ᱮᱫᱟ, ᱟᱨ ᱫᱟᱨᱟᱭ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱫᱚ (n + 1) ᱮᱞ ᱯᱷᱤᱵᱚᱱᱟᱪᱤ ᱮᱞ ᱠᱟᱱᱟ᱾ ᱵᱚᱞᱢᱟᱱ (Bollman) (2021) ᱫᱚ 1, 2, 3 ᱠᱷᱚᱱ ᱮᱦᱚᱵᱚᱜ ᱫᱷᱟᱨᱟ ᱞᱟᱹᱜᱤᱫ ᱤᱱᱰᱟᱠᱥᱚᱱ ᱛᱮ ᱚᱱᱟ ᱜᱮ ᱨᱮᱡᱟᱞᱴ ᱥᱟᱵᱩᱫ ᱮᱫᱟᱭ, ᱡᱟᱦᱟᱸ ᱫᱚ ᱡᱚᱛᱚ ᱮᱞ ᱢᱤᱫ ᱡᱟᱭᱜᱟ ᱥᱟᱨᱮᱡ ᱮᱫᱟ᱾ 1-ᱭᱩᱱᱤᱴ ᱮᱦᱚᱵ ᱠᱷᱚᱱ 500-ᱭᱩᱱᱤᱴ ᱥᱤᱢᱟᱹ ᱨᱮ, ᱢᱟᱨᱴᱤᱝᱜᱮᱞ ᱫᱚ ᱟᱡᱟᱜ 10 ᱮᱞ ᱵᱟᱡᱤ, 512 ᱭᱩᱱᱤᱴ, ᱵᱟᱝ ᱞᱟᱜᱟᱣ ᱫᱟᱲᱮᱭᱟᱜ-ᱟ᱾ ᱯᱷᱤᱵᱚᱱᱟᱪᱤ ᱫᱚ ᱟᱡᱟᱜ 14 ᱮᱞ, 377 ᱭᱩᱱᱤᱴ, ᱞᱟᱜᱟᱣ ᱫᱟᱲᱮᱭᱟᱜ-ᱟ, ᱢᱮᱱᱠᱷᱟᱱ 15 ᱮᱞ, 610, ᱫᱚ ᱵᱟᱝ᱾
ᱢᱤᱫ ᱰᱚᱡᱮᱱ ᱥᱮ ᱠᱚᱞᱚᱢ ᱨᱮ ᱪᱮᱫ ᱵᱚᱫᱚᱞᱚᱜ-ᱟ?
ᱰᱚᱡᱮᱱᱵᱟᱡᱤᱰᱚᱡᱮᱱ1–12, 13–24 ᱟᱨᱵᱟᱝ 25–36, ᱵᱷᱩᱜᱛᱟᱱ 2:1᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱥᱮ ᱠᱚᱞᱚᱢᱵᱟᱡᱤᱠᱚᱞᱚᱢᱞᱮᱭᱟᱣᱴ ᱨᱮᱭᱟᱜ ᱢᱤᱫ ᱛᱷᱤᱨ ᱠᱚᱞᱚᱢ ᱨᱮ ᱜᱮᱞᱵᱟᱨ ᱮᱞ, ᱵᱷᱩᱜᱛᱟᱱ 2:1᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱨᱮ, ᱡᱟᱦᱟᱸ ᱫᱚ 2 ᱨᱮ 1 ᱮᱢ ᱮᱫᱟ, ᱛᱟᱭᱚᱢ ᱫᱷᱟᱯ ᱵᱟᱝ ᱞᱟᱹᱠᱛᱤᱭᱟ᱾ F ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱨᱮ ᱡᱤᱛ ᱫᱚ 2F ᱮᱢ ᱮᱫᱟ, ᱡᱟᱦᱟᱸ ᱫᱚ ᱥᱟᱵᱟᱦ ᱞᱟᱦᱟ ᱨᱮᱭᱟᱜ ᱡᱚᱛᱚ ᱦᱟᱹᱱᱤ ᱠᱷᱚᱱ ᱵᱮᱥᱤ ᱜᱮᱭᱟ, ᱚᱱᱟᱛᱮ ᱡᱚᱛᱚ ᱡᱤᱛ ᱫᱚ ᱫᱷᱟᱨᱟ ᱞᱟᱵᱷ ᱥᱟᱶ ᱢᱩᱪᱟᱹᱫ ᱮᱫᱟ᱾ ᱫᱟᱢ ᱫᱚ ᱱᱚᱶᱟ ᱠᱟᱱᱟ ᱡᱮ ᱢᱤᱫ ᱰᱚᱡᱮᱱ ᱫᱚ 37 ᱨᱮ 12 ᱫᱷᱟᱣ ᱜᱮ ᱡᱤᱛᱚᱜ-ᱟ, ᱚᱱᱟᱛᱮ ᱡᱮᱞᱮᱧ ᱦᱟᱹᱱᱤ ᱞᱟᱜᱟᱛᱟᱨ ᱫᱚ ᱟᱹᱰᱤ ᱵᱮᱥᱤ ᱦᱩᱭᱩᱜ-ᱟ: ᱞᱟᱜᱟᱛᱟᱨ ᱜᱮᱞ ᱯᱩᱱ ᱰᱚᱡᱮᱱ ᱦᱟᱹᱱᱤ ᱫᱚ 242 ᱪᱮᱥᱴᱟ ᱨᱮ ᱢᱤᱫ ᱫᱷᱟᱣ ᱞᱮᱠᱟ ᱦᱩᱭᱩᱜ-ᱟ, ᱟᱨᱟᱜᱵᱟᱡᱤᱟᱨᱟᱜ18 ᱟᱨᱟᱜ ᱮᱞ ᱨᱮ ᱥᱚᱢᱟᱱ ᱴᱟᱠᱟ ᱵᱟᱡᱤ᱾ ᱯᱷᱨᱮᱧᱪ ᱨᱮ: ᱨᱩᱡᱽ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱞᱟᱹᱜᱤᱫ 11,000 ᱨᱮ ᱢᱤᱫ ᱫᱷᱟᱣ ᱞᱮᱠᱟ ᱛᱩᱞᱟᱹ᱾
ᱞᱟᱜᱟᱛᱟᱨ ᱵᱟᱨ ᱡᱤᱛ ᱛᱟᱭᱚᱢ ᱜᱮ ᱛᱷᱤᱨ ᱞᱮᱠᱷᱟᱱ ᱪᱮᱫ ᱦᱩᱭᱩᱜ-ᱟ?
ᱮᱴᱟᱜ ᱢᱤᱫ ᱨᱩᱯ ᱫᱚ ᱡᱚᱛᱚ ᱜᱷᱩᱨᱟᱹᱣ ᱛᱟᱭᱚᱢ, ᱡᱤᱛ ᱥᱮ ᱦᱟᱹᱱᱤ, ᱢᱤᱫ ᱫᱷᱟᱯ ᱢᱟᱲᱟᱝ ᱥᱟᱨᱮᱡ ᱮᱫᱟ, ᱟᱨ ᱞᱟᱜᱟᱛᱟᱨ ᱵᱟᱨ ᱡᱤᱛ ᱛᱟᱭᱚᱢ ᱜᱮ ᱛᱷᱤᱨᱚᱜ-ᱟ᱾ ᱵᱟᱨ ᱡᱤᱛ ᱞᱟᱦᱟ n ᱦᱟᱹᱱᱤ ᱥᱟᱶ, ᱞᱟᱵᱷ ᱫᱚ (n + 1) ᱮᱞ ᱯᱷᱤᱵᱚᱱᱟᱪᱤ ᱮᱞ ᱥᱟᱶ 1 ᱠᱟᱱᱟ: ᱡᱤᱛ ᱞᱚᱜᱚᱱ ᱦᱮᱡ ᱞᱮᱠᱷᱟᱱ ᱦᱩᱰᱤᱧ, ᱵᱤᱞᱚᱢ ᱦᱮᱡ ᱞᱮᱠᱷᱟᱱ ᱢᱟᱨᱟᱝ, ᱟᱨ ᱚᱱᱟ ᱞᱮᱠᱟ ᱜᱮ ᱵᱟᱲᱦᱟᱣ ᱟᱠᱟᱱ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱨᱮ ᱡᱤᱛᱚᱜ-ᱟ᱾ ᱞᱚᱜᱚᱱ ᱵᱟᱨ ᱡᱤᱛ ᱫᱚ ᱥᱟᱵᱟᱦ ᱦᱩᱰᱤᱧ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱨᱮ ᱦᱮᱡᱚᱜ-ᱟ; ᱵᱤᱞᱚᱢ ᱵᱟᱨ ᱡᱤᱛ ᱞᱟᱹᱜᱤᱫ ᱫᱚ ᱴᱷᱤᱠ ᱚᱱᱟ ᱚᱠᱛᱚ ᱨᱮ ᱢᱤᱫ ᱢᱟᱨᱟᱝ ᱵᱟᱡᱤ ᱞᱟᱹᱠᱛᱤᱭᱟ ᱡᱚᱠᱷᱚᱱ ᱥᱮᱥᱚᱱ ᱫᱚ ᱡᱟᱹᱥᱛᱤ ᱠᱷᱟᱨᱟᱯ ᱦᱟᱹᱞᱚᱛ ᱨᱮ ᱢᱮᱱᱟᱜ-ᱟ᱾
ᱥᱤᱢᱩᱞᱮᱴᱚᱨ · ᱯᱷᱤᱵᱚᱱᱟᱪᱤ
$1,000 ᱠᱷᱮᱞ ᱯᱩᱸᱡᱤ ᱠᱷᱚᱱ ᱥᱚᱢᱟᱱ ᱴᱟᱠᱟ ᱵᱟᱡᱤ ᱨᱮᱭᱟᱜ 2,000 ᱥᱮᱥᱚᱱ᱾
ᱢᱩᱞ ᱵᱟᱡᱤ $10 · ᱴᱮᱵᱩᱞ ᱥᱤᱢᱟᱹ $500 · ᱢᱤᱫ ᱥᱮᱥᱚᱱ ᱨᱮ 100 ᱜᱷᱩᱨᱟᱹᱣ
ᱦᱟᱹᱱᱤ ᱨᱮᱭᱟᱜ ᱞᱟᱜᱟᱛᱟᱨ ᱪᱮᱫ ᱥᱟᱦᱟᱣ ᱦᱤᱡᱩᱜ-ᱟ
ᱥᱤᱝᱜᱚᱞ ᱡᱤᱨᱚ ᱪᱟᱠᱟ ᱨᱮ ᱢᱤᱫ ᱥᱚᱢᱟᱱ ᱴᱟᱠᱟ ᱵᱟᱡᱤ ᱫᱚ 37 ᱜᱷᱩᱨᱟᱹᱣ ᱨᱮ 19 ᱫᱷᱟᱣ ᱦᱟᱨᱟᱣᱚᱜ-ᱟ᱾ ᱢᱤᱫ ᱢᱤᱫ ᱛᱷᱟᱨ ᱫᱚ ᱩᱫᱩᱜ ᱮᱫᱟ ᱡᱮ ᱢᱤᱫ ᱜᱟᱱ ᱜᱷᱩᱨᱟᱹᱣ ᱡᱚᱛᱚ ᱦᱟᱨᱟᱣ ᱨᱮᱭᱟᱜ ᱥᱚᱢᱵᱷᱟᱵᱱᱟ ᱪᱮᱫ, ᱟᱨ ᱡᱚᱛᱚ ᱦᱟᱹᱱᱤ ᱛᱟᱭᱚᱢ ᱥᱤᱨᱤᱡᱽ ᱨᱮ ᱢᱤᱫ ᱫᱷᱟᱯ ᱥᱟᱢᱟᱝ ᱪᱟᱞᱟᱜ ᱠᱟᱫ ᱢᱤᱫ ᱯᱷᱤᱵᱚᱱᱟᱪᱤ ᱠᱷᱮᱞᱚᱸᱰ ᱚᱱᱟ ᱦᱟᱹᱵᱤᱡ ᱛᱤᱱᱟᱹ ᱦᱟᱨᱟᱣ ᱟᱠᱟᱫᱟᱭ᱾
| ᱞᱟᱜᱟᱛᱟᱨ ᱦᱟᱹᱱᱤ | ᱥᱚᱢᱵᱷᱟᱵᱱᱟ | ᱢᱚᱴᱟ ᱢᱩᱴᱤ 1 / | ᱱᱤᱛᱚᱜ ᱦᱟᱹᱵᱤᱡ ᱦᱟᱹᱱᱤ | ᱫᱟᱨᱟᱭ ᱵᱟᱡᱤ |
|---|---|---|---|---|
| 1 | 51.4% | 2 | $1 | $1 |
| 2 | 26.4% | 4 | $2 | $2 |
| 3 | 13.5% | 7 | $4 | $3 |
| 4 | 7.0% | 14 | $7 | $5 |
| 5 | 3.6% | 28 | $12 | $8 |
| 6 | 1.8% | 55 | $20 | $13 |
| 7 | 0.94% | 106 | $33 | $21 |
| 8 | 0.48% | 207 | $54 | $34 |
| 9 | 0.25% | 403 | $88 | $55 |
| 10 | 0.13% | 784 | $143 | $89 |
| 11 | 0.065% | 1527 | $232 | $144 |
| 12 | 0.034% | 2974 | $376 | $233 |
| 13 | 0.017% | 5792 | $609 | $377 |
| 14 | 0.0089% | 11279 | $986 | $610 |
| 15 | 0.0046% | 21965 | $1,596 | $987 |
ᱥᱚᱢᱵᱷᱟᱵᱱᱟ ᱫᱚ 19/37 ᱠᱟᱱᱟ, ᱢᱤᱫ ᱢᱤᱫ ᱦᱟᱹᱱᱤ ᱨᱮ ᱟᱡ ᱥᱟᱶ ᱜᱩᱱ ᱠᱟᱛᱮ, ᱢᱤᱫ ᱛᱷᱤᱨ ᱜᱷᱩᱨᱟᱹᱣ ᱠᱷᱚᱱ ᱞᱮᱠᱷᱟ ᱠᱟᱛᱮ᱾ ᱢᱤᱫ ᱡᱮᱞᱮᱧ ᱥᱮᱥᱚᱱ ᱨᱮ ᱟᱭᱢᱟ ᱜᱷᱩᱨᱟᱹᱣ ᱢᱮᱱᱟᱜ-ᱟ ᱡᱟᱦᱟᱸ ᱠᱷᱚᱱ ᱱᱚᱶᱟ ᱞᱮᱠᱟᱱ ᱞᱟᱜᱟᱛᱟᱨ ᱮᱦᱚᱵ ᱫᱟᱲᱮᱭᱟᱜ-ᱟ, ᱚᱱᱟᱛᱮ ᱢᱤᱫ ᱟᱭᱩᱵ ᱨᱮ ᱱᱚᱣᱟ ᱫᱚ ᱢᱤᱫ ᱛᱷᱟᱨ ᱩᱫᱩᱜ ᱠᱷᱚᱱ ᱟᱹᱰᱤ ᱵᱮᱥᱤ ᱦᱩᱭ ᱫᱟᱲᱮᱭᱟᱜ-ᱟ᱾
ᱠᱩᱠᱞᱤ ᱠᱚ
ᱦᱮᱸ: ᱡᱤᱛ ᱛᱟᱭᱚᱢ ᱢᱟᱲᱟᱝ, ᱦᱟᱹᱱᱤ ᱛᱟᱭᱚᱢ ᱛᱟᱭᱚᱢ᱾ ᱱᱚᱶᱟ ᱫᱚ ᱢᱤᱫ ᱞᱟᱦᱟᱭ ᱯᱟᱨᱚᱞᱤᱥᱴᱨᱮᱴᱮᱡᱤᱯᱟᱨᱚᱞᱤᱯᱚᱡᱤᱴᱤᱵᱷ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ: ᱡᱤᱛ ᱛᱟᱭᱚᱢ ᱰᱚᱵᱚᱞ, ᱦᱟᱨᱟᱣ ᱛᱟᱭᱚᱢ ᱟᱨᱵᱟᱝ ᱞᱟᱜᱟᱛᱟᱨ ᱯᱮ ᱡᱤᱛ ᱛᱟᱭᱚᱢ ᱫᱚᱦᱲᱟ ᱮᱛᱚᱦᱚᱵ᱾ ᱮᱴᱟᱜ ᱧᱩᱛᱩᱢ: ᱩᱞᱴᱟᱹ ᱢᱟᱨᱴᱤᱝᱜᱮᱞ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱞᱮᱠᱟ ᱪᱟᱞᱟᱜ-ᱟ᱾
ᱜᱚᱱᱤᱛ ᱥᱟᱶ ᱨᱟᱹᱥᱠᱟᱹ
ᱱᱚᱶᱟ ᱠᱚ ᱯᱩᱭᱞᱩ ᱟᱯᱮ ᱛᱮ ᱠᱩᱨᱩᱢᱩᱴᱩ ᱯᱮ᱾ ᱢᱤᱫ ᱢᱤᱫ ᱥᱚᱢᱟᱫᱷᱟᱱ ᱫᱚ ᱩᱠᱩ ᱢᱮᱱᱟᱜ-ᱟ; ᱛᱮᱭᱟᱨ ᱦᱩᱭ ᱠᱟᱛᱮ ᱡᱷᱤᱡᱽ ᱯᱮ᱾
ᱚᱱᱟ ᱨᱩᱯ ᱠᱷᱮᱞ ᱯᱮ ᱡᱟᱦᱟᱸ ᱫᱚ ᱡᱚᱛᱚ ᱜᱷᱩᱨᱟᱹᱣ ᱛᱟᱭᱚᱢ, ᱡᱤᱛ ᱥᱮ ᱦᱟᱹᱱᱤ, ᱥᱤᱨᱤᱡᱽ ᱨᱮ ᱢᱤᱫ ᱫᱷᱟᱯ ᱥᱮᱱᱚᱜ-ᱟ, ᱟᱨ ᱞᱟᱜᱟᱛᱟᱨ ᱵᱟᱨ ᱡᱤᱛ ᱛᱟᱭᱚᱢ ᱛᱤᱝᱜᱩᱜ-ᱟ: ᱵᱟᱡᱤ ᱴᱟᱠᱟ 1, 1, 2, 3, 5, 8 ᱟᱨ ᱚᱱᱟ ᱞᱮᱠᱟ᱾ ᱯᱟᱹᱦᱤᱞ ᱜᱮᱞ ᱯᱩᱱ ᱨᱮᱡᱟᱞᱴ ᱫᱚ ᱡᱤᱛ ᱦᱟᱹᱱᱤ ᱡᱤᱛ ᱦᱟᱹᱱᱤ ᱦᱟᱹᱱᱤ ᱦᱟᱹᱱᱤ ᱡᱤᱛ ᱦᱟᱹᱱᱤ ᱦᱟᱹᱱᱤ ᱦᱟᱹᱱᱤ ᱡᱤᱛ ᱦᱟᱹᱱᱤ ᱦᱟᱹᱱᱤ ᱡᱤᱛ᱾ ᱟᱯᱮ ᱚᱠᱟ ᱨᱮ ᱢᱮᱱᱟᱢ ᱯᱮ, ᱟᱨ ᱫᱟᱨᱟᱭ ᱵᱟᱡᱤ ᱪᱮᱫ?
ᱥᱚᱢᱟᱫᱷᱟᱱ ᱩᱫᱩᱜ
ᱛᱮᱞᱟ 22 ᱭᱩᱱᱤᱴᱴᱟᱠᱟᱭᱩᱱᱤᱴᱢᱤᱫ ᱥᱤᱥᱴᱮᱢ ᱟᱨᱵᱟᱝ ᱠᱷᱮᱞᱚᱸᱰ ᱵᱮᱵᱷᱟᱨ ᱵᱮᱥ ᱵᱟᱡᱤ ᱴᱟᱠᱟ᱾ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ ᱠᱚ ᱭᱩᱱᱤᱴ ᱛᱮ ᱵᱩᱡᱷᱟᱹᱣᱚᱜ-ᱟ ᱡᱮᱢᱚᱱ ᱡᱟᱦᱟᱸᱱ ᱴᱮᱵᱩᱞ ᱨᱮ ᱞᱟᱹᱜᱩ ᱦᱩᱭᱩᱜ-ᱟ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱦᱟᱹᱱᱤ᱾ ᱫᱟᱨᱟᱭ ᱵᱟᱡᱤ ᱫᱚ 610 ᱭᱩᱱᱤᱴ, ᱟᱨ ᱢᱤᱫ ᱡᱤᱛ ᱥᱮᱥᱚᱱ ᱫᱚ 588 ᱞᱟᱵᱷ ᱨᱮ ᱢᱩᱪᱟᱹᱫᱟ᱾
ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱠᱚ ᱫᱚ ᱯᱟᱹᱦᱤᱞ ᱜᱮᱞ ᱯᱩᱱ ᱯᱷᱤᱵᱚᱱᱟᱪᱤᱥᱴᱨᱮᱴᱮᱡᱤᱯᱷᱤᱵᱚᱱᱟᱪᱤ1, 1, 2, 3, 5, 8, 13… ᱥᱤᱨᱤᱡᱽ ᱯᱟᱧᱡᱟ ᱯᱨᱚᱜᱽᱨᱮᱥᱚᱱ: ᱦᱟᱨᱟᱣ ᱛᱟᱭᱚᱢ ᱢᱤᱫ ᱫᱷᱟᱯ ᱞᱟᱦᱟ, ᱡᱤᱛ ᱛᱟᱭᱚᱢ ᱵᱟᱨ ᱫᱷᱟᱯ ᱛᱟᱭᱚᱢ᱾ᱯᱩᱨᱟᱹ ᱯᱟᱲᱦᱟᱣ → ᱮᱞ ᱠᱟᱱᱟ: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 ᱟᱨ 377᱾ ᱡᱤᱛ ᱠᱚ ᱥᱮᱞᱮᱫ ᱯᱮ ᱟᱨ ᱦᱟᱹᱱᱤ ᱠᱚ ᱚᱪᱚᱜ ᱯᱮ: +1 −1 +2 −3 −5 −8 +13 −21 −34 −55 +89 −144 −233 +377 = −22᱾
ᱢᱩᱪᱟᱹᱫ ᱨᱮᱡᱟᱞᱴ ᱫᱚ ᱡᱤᱛ ᱛᱟᱦᱮᱸ ᱠᱟᱱᱟ, ᱚᱱᱟᱛᱮ ᱟᱨᱦᱚᱸ ᱢᱤᱫ ᱡᱤᱛ ᱛᱮ ᱢᱩᱪᱟᱹᱫᱚᱜ-ᱟ᱾ ᱫᱟᱨᱟᱭ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱫᱚ 610: ᱡᱤᱛ ᱞᱮᱠᱷᱟᱱ +588, ᱦᱟᱹᱱᱤ ᱞᱮᱠᱷᱟᱱ −632 ᱟᱨ ᱫᱟᱨᱟᱭ ᱵᱟᱡᱤ ᱴᱟᱠᱟ 987᱾
ᱱᱚᱣᱟ ᱨᱩᱯ ᱫᱚ ᱡᱟᱦᱟᱸ ᱚᱠᱛᱚ ᱦᱚᱸ ᱞᱟᱵᱷ ᱨᱮ ᱢᱩᱪᱟᱹᱫᱚᱜ-ᱟ, ᱢᱮᱱᱠᱷᱟᱱ ᱱᱚᱰᱮ ᱚᱱᱟ ᱞᱟᱹᱜᱤᱫ 610 ᱭᱩᱱᱤᱴ ᱵᱟᱡᱤ ᱞᱟᱹᱠᱛᱤᱭᱟ᱾ ᱥᱟᱫᱷᱟᱨᱚᱬ ᱱᱤᱭᱚᱢ ᱨᱮ, ᱡᱤᱛ ᱛᱟᱭᱚᱢ ᱵᱟᱨ ᱫᱷᱟᱯ ᱛᱟᱭᱚᱢ, ᱚᱱᱟ ᱜᱮ ᱜᱮᱞ ᱯᱩᱱ ᱨᱮᱡᱟᱞᱴ ᱫᱚ ᱮᱠᱮᱱ 1 ᱭᱩᱱᱤᱴ ᱦᱟᱹᱱᱤ ᱟᱨ ᱫᱟᱨᱟᱭ ᱵᱟᱡᱤ 2 ᱵᱟᱹᱜᱤᱭᱟ᱾ ᱵᱚᱞᱢᱟᱱ (Bollman) (2021), ᱡᱟᱦᱟᱸᱭ ᱥᱤᱨᱤᱡᱽ 1, 2, 3 ᱠᱷᱚᱱ ᱮᱦᱚᱵᱟ, ᱩᱱᱤ ᱫᱚ 1 ᱦᱟᱹᱱᱤ ᱟᱨ ᱫᱟᱨᱟᱭ ᱵᱟᱡᱤ 3 ᱧᱟᱢᱟᱭ᱾ ᱱᱚᱰᱮ ᱥᱟᱫᱷᱟᱨᱚᱬ ᱱᱤᱭᱚᱢ ᱫᱚ ᱠᱚᱢ ᱦᱟᱨᱟᱣᱟ ᱪᱮᱫᱟᱜ ᱥᱮ ᱡᱚᱛᱚ ᱡᱤᱛ ᱫᱚ ᱵᱟᱡᱤ ᱴᱟᱠᱟ ᱠᱚ ᱛᱟᱭᱚᱢ ᱞᱟᱛᱟᱨ ᱛᱮ ᱟᱹᱜᱩᱭᱟ᱾
ᱜᱷᱩᱨᱟᱹᱣᱵᱟᱡᱤ ᱴᱟᱠᱟᱨᱮᱡᱟᱞᱴᱪᱟᱞᱟᱜ ᱡᱚᱛᱚ11ᱡᱤᱛ+121ᱦᱟᱹᱱᱤ032ᱡᱤᱛ+243ᱦᱟᱹᱱᱤ−155ᱦᱟᱹᱱᱤ−668ᱦᱟᱹᱱᱤ−14713ᱡᱤᱛ−1821ᱦᱟᱹᱱᱤ−22934ᱦᱟᱹᱱᱤ−561055ᱦᱟᱹᱱᱤ−1111189ᱡᱤᱛ−2212144ᱦᱟᱹᱱᱤ−16613233ᱦᱟᱹᱱᱤ−39914377ᱡᱤᱛ−22ᱚᱱᱟ ᱜᱮ ᱨᱩᱯ ᱨᱮ, n ᱦᱟᱹᱱᱤ ᱛᱟᱭᱚᱢ ᱵᱟᱨ ᱡᱤᱛ ᱫᱚ ᱪᱮᱫᱟᱜ ᱡᱟᱦᱟᱸ ᱚᱠᱛᱚ ᱦᱚᱸ (n + 1) ᱥᱟᱭ ᱯᱷᱤᱵᱚᱱᱟᱪᱤ ᱮᱞ ᱥᱮᱞᱮᱫ 1 ᱞᱟᱵᱷ ᱵᱟᱹᱜᱤᱭᱟ?
ᱥᱚᱢᱟᱫᱷᱟᱱ ᱩᱫᱩᱜ
ᱛᱮᱞᱟ n ᱦᱟᱹᱱᱤ ᱫᱚ ᱡᱩᱲᱤ ᱠᱟᱛᱮ (n + 2) ᱥᱟᱭ ᱯᱷᱤᱵᱚᱱᱟᱪᱤ ᱮᱞ ᱠᱷᱚᱱ 1 ᱠᱚᱢ ᱦᱩᱭᱩᱜ-ᱟ, ᱟᱨ ᱵᱟᱨ ᱡᱤᱛ ᱫᱚ ᱫᱟᱨᱟᱭ ᱵᱟᱨ ᱮᱞ ᱠᱟᱱᱟ, ᱡᱟᱦᱟᱸ ᱫᱚ ᱢᱤᱫ ᱠᱟᱛᱮ ᱚᱱᱟ ᱠᱷᱚᱱ (n + 1) ᱥᱟᱭ ᱮᱞ ᱥᱮᱞᱮᱫ 1 ᱵᱟᱹᱲᱛᱤ᱾
1, 1, 2, 3, 5 … ᱞᱟᱹᱜᱤᱫ F₁, F₂, F₃ … ᱚᱞ ᱯᱮ᱾ ᱡᱚᱛᱚ ᱮᱞ ᱫᱚ ᱚᱱᱟ ᱞᱟᱹᱦᱟ ᱨᱮᱭᱟᱜ ᱵᱟᱨ ᱮᱞ ᱨᱮᱭᱟᱜ ᱡᱚᱛᱚ ᱠᱟᱱᱟ᱾
ᱦᱟᱹᱱᱤ ᱠᱚ: F₁ + F₂ + … + Fₙ᱾ ᱚᱱᱟ ᱡᱚᱛᱚ ᱨᱮ 1 ᱥᱮᱞᱮᱫ ᱯᱮ, F₂ ᱞᱮᱠᱟᱛᱮ ᱚᱞ ᱠᱟᱛᱮ, ᱟᱨ ᱚᱱᱟ ᱦᱩᱰᱤᱧᱚᱜ-ᱟ: F₂ + F₁ = F₃, ᱮᱱᱟᱛᱮ F₃ + F₂ = F₄, ᱟᱨ ᱚᱱᱟ ᱞᱮᱠᱟ, ᱯᱩᱨᱟᱹ ᱡᱚᱛᱚ Fₙ₊₂ ᱦᱩᱭᱩᱜ ᱦᱟᱹᱵᱤᱡ᱾ ᱚᱱᱟᱛᱮ ᱦᱟᱹᱱᱤ ᱨᱮᱭᱟᱜ ᱡᱚᱛᱚ Fₙ₊₂ − 1᱾
ᱡᱤᱛ ᱠᱚ ᱫᱚ ᱫᱟᱨᱟᱭ ᱵᱟᱨ ᱵᱟᱡᱤ ᱴᱟᱠᱟ, Fₙ₊₁ ᱟᱨ Fₙ₊₂᱾ ᱞᱟᱵᱷ: Fₙ₊₁ + Fₙ₊₂ − (Fₙ₊₂ − 1) = Fₙ₊₁ + 1᱾ ᱯᱩᱱ ᱦᱟᱹᱱᱤ ᱥᱟᱶ, ᱚᱱᱟ ᱫᱚ F₅ + 1 = 6 ᱭᱩᱱᱤᱴ᱾
ᱞᱟᱸᱫᱟ ᱞᱟᱹᱜᱤᱫ ᱠᱚᱢᱤᱠ
ᱥᱚᱱᱟ ᱟᱹᱱᱩᱯᱟᱹᱛ ᱫᱚ ᱵᱮᱥᱤ ᱥᱮ ᱵᱮᱥᱤ ᱵᱟᱡᱤ ᱨᱮᱭᱟᱜ ᱵᱚᱨᱰ ᱥᱟᱶ ᱧᱟᱯᱟᱢᱚᱜ-ᱟ᱾
ᱯᱷᱮᱰᱟᱛ
- RouletteDoc. Strategy description and examples.
- Roulette77. Strategy description and examples.
- Ethier, S. N. The Doctrine of Chances: Probabilistic Aspects of Gambling.
- Epstein, R. A. The Theory of Gambling and Statistical Logic, 2nd ed.
- Bollman, M. Mathematics of the Big Four Casino Table Games: Blackjack, Baccarat, Craps, & Roulette.
- Sigler, L. E. Fibonacci’s Liber Abaci: A Translation into Modern English of Leonardo Pisano’s Book of Calculation.
- Singh, P. The so-called Fibonacci numbers in ancient and medieval India.