\u039f\u03b9 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03af Fibonacci \u03ad\u03c7\u03bf\u03c5\u03bd \u03c3\u03b7\u03bc\u03b1\u03bd\u03c4\u03b9\u03ba\u03ad\u03c2 \u03b9\u03b4\u03b9\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2. \u0395\u03b4\u03ce \u03b8\u03b1 \u03b1\u03bd\u03b1\u03c6\u03ad\u03c1\u03bf\u03c5\u03bc\u03b5 \u03bc\u03b5\u03c1\u03b9\u03ba\u03ad\u03c2 \u03b1\u03c0’ \u03b1\u03c5\u03c4\u03ad\u03c2. \u03a0\u03c1\u03ce\u03c4\u03b1 \u03b2\u03ad\u03b2\u03b1\u03b9\u03b1 \u03b8\u03b1 \u03c3\u03c5\u03bc\u03b2\u03bf\u03bb\u03af\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf\u03c5\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd\u03c2 \u03c3\u03c4\u03b7 \u03c3\u03b5\u03b9\u03c1\u03ac. \u0388\u03c4\u03c3\u03b9 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \\(F_1=1,\\; F_2=1, \\; F_3=2, \\; F_4=3, \\; F_5=5,\\; F_6=8…\\) \u03ba.\u03bf.\u03ba
\n1. \u0394\u03cd\u03bf \u03c3\u03c5\u03bd\u03b5\u03c7\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03af Fibonacci \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03c1\u03ce\u03c4\u03bf\u03b9 \u03c0\u03c1\u03bf\u03c2 \u03b1\u03bb\u03bb\u03ae\u03bb\u03bf\u03c5\u03c2, \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03b4\u03b5\u03bd \u03ad\u03c7\u03bf\u03c5\u03bd \u03ba\u03bf\u03b9\u03bd\u03bf\u03cd\u03c2 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03bd\u03c4\u03b5\u03c2 \u03c0\u03b1\u03c1\u03b1 \u03bc\u03cc\u03bd\u03bf \u03c4\u03bf 1.
\n2. \u03a4\u03bf \u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03b1 \u03c4\u03c9\u03bd \\(n\\) \u03c0\u03c1\u03ce\u03c4\u03c9\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd Fibonacci \u03b5\u03af\u03bd\u03b1\u03b9 \\(\\sum_{i=1}^{n}F_i=F_{n+2}-1\\)
\n3. \u03a4\u03bf \u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03b1 \u03c4\u03c9\u03bd \u03c4\u03b5\u03c4\u03c1\u03b1\u03b3\u03ce\u03bd\u03c9\u03bd \u03c4\u03c9\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd Fibonacci \u03b3\u03b9\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b4\u03b5\u03b9\u03b3\u03bc\u03b1 \u03c4\u03c9\u03bd 6 \u03c0\u03c1\u03ce\u03c4\u03c9\u03bd, \u03b5\u03af\u03bd\u03b1\u03b9\u00a0 \u00a0\\(1^2+1^2+2^2+3^2+5^2+8^2 +13^2=273=13\\cdot 21\\) \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03b7 \\[\\sum_{i=1}^{n}F_{i}^2=F_n\\cdot F_{n+1}.\\] \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 \\(n.\\)
\n4. \u0391\u03bd \u03c0\u03ac\u03c1\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf \u03c4\u03b5\u03c4\u03c1\u03ac\u03b3\u03c9\u03bd\u03bf \u03b5\u03bd\u03cc\u03c2 Fibonacci \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd \u03c0.\u03c7. \u03c4\u03bf\u03c5 \\(F_7=13^2=169\\) \u03ba\u03b1\u03b9 \u03c4\u03bf\u03c5 \u03b1\u03c6\u03b1\u03b9\u03c1\u03ad\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf \u03c4\u03b5\u03c4\u03c1\u03ac\u03b3\u03c9\u03bd\u03bf \u03b5\u03bd\u03cc\u03c2 \u03bc\u03b9\u03ba\u03c1\u03cc\u03c4\u03b5\u03c1\u03bf\u03c5 \u03ba\u03b1\u03c4\u03ac \u03b4\u03cd\u03bf \u03b8\u03ad\u03c3\u03b5\u03b9\u03c2 \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03b5\u03b4\u03ce \u03c4\u03bf\u03c5 \\(F_5=5^2=25\\), \u03b8\u03b1 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \\(F_{7}^2-F{5}^2=169-25=144=F_12.\\) \u03a4\u03bf \u03af\u03b4\u03b9\u03bf \u03c3\u03c5\u03bc\u03b2\u03b1\u03af\u03bd\u03b5\u03b9 \u03ba\u03b1\u03b9 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ac \u03c4\u03b5\u03c4\u03c1\u03b1\u03b3\u03ce\u03bd\u03c9\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd Fibonacci \u03c0.\u03c7. \\(F_{15}^2-F_{13}^2=610^2-233^2=372100-54289=317811=F_28.\\) \u0398\u03b1 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03c4\u03b7 \u03c3\u03c7\u03ad\u03c3\u03b7 \\[F_{n}^2-F_{n-2}^2=F_{2n-2}.\\]
\n5. \u0399\u03c3\u03c7\u03cd\u03b5\u03b9 \\(F_{n}^2+F_{n+1}^2=F_{2n+1}.\\)
\n6. \u0399\u03c3\u03c7\u03cd\u03b5\u03b9 \\(F_{n-1}\\cdot F_{n+1}=F_{n}^2+(-1)^n.\\)
\n7. \u039e\u03b5\u03ba\u03b9\u03bd\u03ce\u03bd\u03c4\u03b1\u03c2 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03c0\u03c1\u03ce\u03c4\u03bf \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc Fibonacci \u03c0\u03b1\u03c1\u03b1\u03c4\u03b7\u03c1\u03bf\u03cd\u03bc\u03b5 \u03cc\u03c4\u03b9 \u03ba\u03ac\u03b8\u03b5 \u03b5\u03c0\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03c4\u03c1\u03af\u03c4\u03bf\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03ac\u03c1\u03c4\u03b9\u03bf\u03c2, \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03b4\u03b9\u03b1\u03b9\u03c1\u03b5\u03af\u03c4\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03bf 2 (\u03c0\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf \\(F_3\\), \u03ba\u03ac\u03b8\u03b5 \u03c4\u03ad\u03c4\u03b1\u03c1\u03c4\u03bf\u03c2 \u03b5\u03c0\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03b4\u03b9\u03b1\u03b9\u03c1\u03b5\u03af\u03c4\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03bf 3, \u03c0\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf \\(F_4\\), \u03ba\u03ac\u03b8\u03b5 5\u03bf\u03c2 \u03b5\u03c0\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03b4\u03b9\u03b1\u03b9\u03c1\u03b5\u03af\u03c4\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03bf 5 \u03c0\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf \\(F_5\\). \u0395\u03bb\u03ad\u03b3\u03c7\u03bf\u03bd\u03c4\u03b1\u03c2 \u03b3\u03b9\u03b1 \u03b4\u03b9\u03b1\u03b9\u03c1\u03b5\u03c4\u03cc\u03c4\u03b7\u03c4\u03b1 \u03bc\u03b5 \u03c4\u03bf\u03bd \u03b5\u03c0\u03cc\u03bc\u03b5\u03bd\u03bf \u03b1\u03c1\u03b9\u03b8\u03bc\u03cc Fibonacci, \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03c4\u03bf\u03bd \\(F_6=8\\) \u03b2\u03bb\u03ad\u03c0\u03bf\u03c5\u03bc\u03b5 \u03cc\u03c4\u03b9 \u03bf\u03b9 \\(F_6, F_12, F_18, F_24, F_30…\\) \u03b4\u03b9\u03b1\u03b9\u03c1\u03bf\u03cd\u03bd\u03c4\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03bf 8 \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03c4\u03bf\u03bd \\(F_6.\\) \u0391\u03bd \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03bf \\(k\\) \u03b4\u03b9\u03b1\u03b9\u03c1\u03b5\u03af\u03c4\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03bf \\(m\\), \u03c4\u03cc\u03c4\u03b5 \u03bf \\(F_k\\) \u03b4\u03b9\u03b1\u03b9\u03c1\u03b5\u03af\u03c4\u03b1\u03b9 \u03bc\u03b5 \u03c4\u03bf\u03bd \\(F_m.\\)
\n\u0391\u03c5\u03c4\u03ad\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bc\u03b5\u03c1\u03b9\u03ba\u03ad\u03c2 \u03b1\u03c0\u03cc \u03c4\u03b9\u03c2 \u03b9\u03b4\u03b9\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2 \u03c4\u03c9\u03bd \u03b1\u03c1\u03b9\u03b8\u03bc\u03ce\u03bd \u03c4\u03b7\u03c2 \u03c3\u03b5\u03b9\u03c1\u03ac\u03c2 Fibonacci.<\/p>\n","protected":false},"excerpt":{"rendered":"
\u039f\u03b9 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03af Fibonacci \u03ad\u03c7\u03bf\u03c5\u03bd \u03c3\u03b7\u03bc\u03b1\u03bd\u03c4\u03b9\u03ba\u03ad\u03c2 \u03b9\u03b4\u03b9\u03cc\u03c4\u03b7\u03c4\u03b5\u03c2. \u0395\u03b4\u03ce \u03b8\u03b1 \u03b1\u03bd\u03b1\u03c6\u03ad\u03c1\u03bf\u03c5\u03bc\u03b5 \u03bc\u03b5\u03c1\u03b9\u03ba\u03ad\u03c2 \u03b1\u03c0’ \u03b1\u03c5\u03c4\u03ad\u03c2. \u03a0\u03c1\u03ce\u03c4\u03b1 \u03b2\u03ad\u03b2\u03b1\u03b9\u03b1 \u03b8\u03b1 \u03c3\u03c5\u03bc\u03b2\u03bf\u03bb\u03af\u03c3\u03bf\u03c5\u03bc\u03b5 \u03c4\u03bf\u03c5\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03cd\u03c2 \u03c3\u03c4\u03b7 \u03c3\u03b5\u03b9\u03c1\u03ac. \u0388\u03c4\u03c3\u03b9 \u03ad\u03c7\u03bf\u03c5\u03bc\u03b5 \\(F_1=1,\\; F_2=1, \\; F_3=2, \\; F_4=3, \\; F_5=5,\\; F_6=8…\\) \u03ba.\u03bf.\u03ba 1. \u0394\u03cd\u03bf \u03c3\u03c5\u03bd\u03b5\u03c7\u03cc\u03bc\u03b5\u03bd\u03bf\u03b9 \u03b1\u03c1\u03b9\u03b8\u03bc\u03bf\u03af Fibonacci \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03c1\u03ce\u03c4\u03bf\u03b9 \u03c0\u03c1\u03bf\u03c2 \u03b1\u03bb\u03bb\u03ae\u03bb\u03bf\u03c5\u03c2, \u03b4\u03b7\u03bb\u03b1\u03b4\u03ae \u03b4\u03b5\u03bd \u03ad\u03c7\u03bf\u03c5\u03bd \u03ba\u03bf\u03b9\u03bd\u03bf\u03cd\u03c2 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03bd\u03c4\u03b5\u03c2 \u03c0\u03b1\u03c1\u03b1 \u03bc\u03cc\u03bd\u03bf \u03c4\u03bf 1. 2. \u03a4\u03bf \u03ac\u03b8\u03c1\u03bf\u03b9\u03c3\u03bc\u03b1 \u03c4\u03c9\u03bd \\(n\\) … \u0394\u03b9\u03b1\u03b2\u03ac\u03c3\u03c4\u03b5 \u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03cc\u03c4\u03b5\u03c1\u03b1 “Fibonacci (\u03c3\u03c5\u03bd\u03ad\u03c7\u03b5\u03b9\u03b1)”<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-323","post","type-post","status-publish","format-standard","hentry","category-maths"],"_links":{"self":[{"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/posts\/323","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/comments?post=323"}],"version-history":[{"count":17,"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/posts\/323\/revisions"}],"predecessor-version":[{"id":446,"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/posts\/323\/revisions\/446"}],"wp:attachment":[{"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/media?parent=323"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/categories?post=323"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/maths4u.gr\/index.php\/wp-json\/wp\/v2\/tags?post=323"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}