Ευκλειδης Β 88

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  • - - 2013 3,5

    2013

    "" 2013:

    ...

    4 SEEMOUS 2013

    .

    1 . 1099/96 .

    IMABHMATIIJKH ETAIPEIA

  • ETAIPEIA 88 2013 : 3,50

    e-mail: info@hms.gr www.hms.gr

    .;

    .;

    .f .; Homo Mathematicus

    ' .;

    :

    .; :

    ' . : .;

    :

    .;

    :

    ' .f : .; :

    .;

    .;

    .;

    .; ...

    .; .

    1 8 9

    21

    25 31

    34 40 44

    48 59

    66 72 75 79 71

    , 4 .

    .

    . .

    , .

    :

    : yy ,

    34

    106 79 :

    rrt..: 21 3617784 - 3616532 F: 2103641025 : :

  • ... , -y , -y .

    . Albert Einstein

    . , ...

    . . . . 1.

    . 2.

    . 3. .

    1202 (. 1 1 70-. 1 240) (F ibonacci), Lber avac I 1 ,2,3, . . . ,9 () .

    Guilielmo Bonacci. "filius Bonacci" Bonacci. - Fibonacci "filius Bonacci". , Bigollo "".

    ( ) . .

    529 . ,

    . . ""

    900 " " "" . .

    . , .

    XVI , , .

    529 . 1 500 . . a - (Buggia). , . -

    ' 88 .4/1

  • --------------- . . . ---------------. , , , , .

    , . Liber Abbaci. . / . . ,

    ' 1 4 ' ' ' 2 ,

    4.!_. F ibonacci 2

    / , . .

    F ibonacci . . fn - : h h h h h h h h 2 3 5 8 1 3 2 1 34 55 . . .

    : fn+2 = fn+l + fn n ' fo = 1 'f = 2

    F ibonacci , 2 . - !12 = 377

    :

    Fn+2 = Fn+l + Fn F0 = 1 F1 = 1 F ibonacci , , , 2, 3, 5, 8, 3, 2, 34, 55, 89, 44, 233, 377, 6, 987, ...

    - .

    ---------------------------------------- 1

    2

    ' 88 .4/2

  • --------------- . . . .

    + b -- =- b

    .

    _____ ____ -----

    b----- -v--+b

    +b . -- =-= b

    : b 1 1 +-=-=>1 +-= b

    2 --1 =0 :

    1 +./5 , 1 -./5 =-- =--2 2 ' = -1

    = 1 ,61 8033988749894842 . . . . ' =-0,6 1 80339887 49894842 . . .

    Fibonacci . ( ) . .

    n n+1 (n+ 1)/n 2 3 1,5 3 5 1 ,666666666 ... 5 8 1,6 8 13 1,625 .. . . .. . . . .. . . . . . . . 144 233 1,618055556 . .. 233 377 1,61802575 1 . . . . .. . .. . . .

    .

    .

    Fibonacci ;

    ..,.,---------

    1

    1/2

    2

    .. .. ..

    . .

    . . , .

    = = =

    72 36 .

    2. yv p

    b ( ). -

    ' 88 .4/3

  • ---------------- . . . ( ) . + b .

    h

    +h . , , -

    . "" . . .

    , ,

    3.

    . -

    . .

    p p Fbonacc.

    . . . ( ) ( , "'" .

    Fibonacci. .

    Fibonacci.

    . 88 .4/4

  • --------------- . . .

    4. 6 libonacci 2 - - 1 = - = (1 + .J5)/2 = (1 - .J5)/2 . +' = 1 . ' = -1 .

    2 = + 1 3 = 2 + = ( + 1) + = 2 + 1 4 = 3 + 2 = (2 + 1) + ( + 1) = 3 + 2 5 = 4 + 3 = (3 + 2) + (2 + 1) = s + 3

    ;

    = J5+l =lim Fn =1+----2 n->oo F n-1 }+ --- 1+--1 1+-

    , .

    =l+Jl+l+ , .

    5. t) :: .,/; Pascal

    . - . :

    I ; I / I

    I I I I I 1 I I I I

    I

    to 6 1 I

    I I )5 s 21 7 1

    I I

    1 I 70 56 28 8 1 I I I

    I I I

    . .

    2.

    3. l . 2 .

    4. 1 2 3.

    5. , 3 , 1 5.

    6. 1 , 4, 3 8.

    7. , 5, 6, 1 3.

    8. 1 , 6, , 4, 2 .

    9. 1 , 7, 15 , 10, 1 34. Pascal.

    ' 88 .4/5

  • --------------- . . . --------------

    1 2 3 4 5 6 7 8 9 10 1

    1 1 1 2 1

    1 3 3 1 1 4 6 4 1

    1 5 10 10 5 1 6 15 20

    1 7 21 1 8

    1 2 3 5 8 13 2 34 55

    F ibonacci . ( F ibonacci. Pascal. . 1 .

    F ibonacci. ;

    1 1. 1. 1, 2, 3, 4, S, . . . H , 3, 6, 10, 15,. 5. .

    F ibonacci . . . F ibonacci: , 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 8 9, 144,233, 377, 61 , 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657,75925, 121393, 196418, 317811, . . .

    ( ) . , 1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, , 7, 7, 4, 1, 5, 6, , 7 , 5, 3, 8, 1, ... 150 F ibonacci. 150

    60 F ibonacci . 60.

    . , 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 75925, 121393, 1964 18, 3178 11, . . . 600 F ibonacci 2 300. , , .

    . 3 ; 3, 4, 5 . 3 . 4 F ibonacci. , , 1, 2, 3, 4 . b .

    " :

    : 1.

    . 23 = 6 .

    2. : 26 = 1 2 .

    3. . 5 = 5. .

    4. . 22 = 4 , 3 2 = 9 . 13. 13.

    5, 12, 13

    ' 88 .4/6

  • ---------------------------------- . . . "" F ibonacci 2, 3, 5, 8. . , , 30, 1 6, 34. , -

    6. Fibonacci l 000 1 999 F ibonacci - .

    . 1299 - - . / - , - - - . " - - ;" - . - B iggolo ( - ) - . . 300 - - . - . -

    F ibonacci - . Liber Abaci / , . 1 , 2, 3, 4, 5, 6, 7, 8, 9, 0.

    .

    1, s, 10, L 50, c 100, D 500, = 1000

    , . . . . , . . - .

    Fibonacci : 1. Lber Abbac ( ), 1 202 2. Practca Geometrae ( ), 1220 3. Lber Quadratorum ( ),

    1225. 4. Flos ( ), 1 225.

    5. D mnor gusa ( ).

    . . 6. Commentary on Book of Eucld's elements. -

    . Boyer ' history of Mathematics : " 900 .

    Fibonacci 1240 20 " , "

    . Fibonacci; ; ; ; (

    :

    Fibonacci JA VA.

    Takis. Papachristou@gmail.com

    ' 88 .417

  • ;

    ;

    . :; ; . ' .

    , Cambridge The Hub Eents 1 : ; , . :. : : ; ; ;

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    . ,

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    . a .

    :. . , : (MEG scanners) . , . .

    ' 88 .4/8

  • a yv tJa

    ...

    30 ' "

    23 2013 1 ( ) , = 1, 2, 3, ...

    1 = 2 . = ( + 1) ( 1 + 2 + + ._1 ) , 2. -1 2013 (1 ) : 1 = 2, 2 = 1 = 3 2, 3 =

    4 ( 1 + 2 ) = 4 4 2 = 4 22 ,

    1 2 2 5( ) 5 3 6 ( ) 6 4 4 =- 1 + 2 + 3 = - 24 = 5 2 5 = - 1 + 2 + 3 + 4 =- 64 = 6 2 . 3 3 4 4

    n = ( n + 1 ) 2n- , n = 1, 2, 3, . . . , k .

    n = k + 1 , : k+ = ( k + 2) 2k . k + 2 ( ) k + 2 ( 2 ( ) k-1 ) , k+ = -k- ,+ +G:J ++k = -k- 2+32 +42 ++ k+1 2 k + 2 ( 2 ( ) k-1 ) : k+ =-k- 22 + 32 + 42 ++ k + 1 2

    ( 1 ) 2,

    2 = k+ 2 (21 + 322 +423 + + (k+ 1)2k ) k+l k ' ( 1 ) (2) , :

    k + 2 ( 2 3 k- 1 ( ) k ) k+ =-k- -2 - 2 - 2 - 2 +- 2 + k+ 1 2

    ( 1 )

    (2)

    a,,, k;2 { -1-1;::_; + (k+ 1)2'} k;2 ( -2' + (k+ 1)2' ) (k + 2)2'. : 20 1 3 = 2014 22

    0 12 2

    ( n + 1 ) n = - ( + 2 ++ n- ) , n 2, n-1

    ( n + 2) n+l = -n-( + 2 + . . . + n ) ' n 1,

    ' 88 . 4/9

    (3)

    (4)

    (5)

  • -------- - --------1 + 2 + + n = (-n-)n+I, n;:: . (6)

    n+2

    ( 5) ( 6) n = (-n-)n+

    I - ( n - 1 )n => n+ = ( 2(n + 2))n, n;:: 1 (7) n+ 2 n + 1 n + 1

    n = ( 2(nn

    + 1) )n

    -

    ] =( 2(nn

    + 1) )(

    n

    21)n_2 = . . .

    = ( 2(nn

    + 1 ) )(n

    21) . . (

    2 4) ( 23 ) = ( n + 1) . 2n-2. = ( n + 1) 2n-I' ' ' 2 ' 2014 220 12 1 = . 2013 = . 2 : y = 22 + 5xy + 3y2