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ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
ɑɢɫɥɚ ɚ0, ɚ1, … ɧɚɡɵɜɚɸɬ ɧɟɩɨɥɧɵɦɢ ɱɚɫɬɧɵɦɢ ɢ ɫɱɢɬɚɸɬ, ɱɬɨ ɚ0 – ɰɟɥɨɟ ɱɢɫɥɨ, ɚ ɚ
ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɨɬɪɢɰɚɬɟɥɶɧɵɯ ɚ
, ɚ2, … – ɧɚɬɭɪɚɥɶɧɵɟ (ɯɨɬɹ ɢ ɧɟɨɛɹɡɚɬɟɥɶɧɨ, ɧɨ ɦɵ ɧɟ ɛɭɞɟɦ
1
, ɚ1, …). ɑɢɫɥɚ
0
1
aa
,
,
DDD
0100
a
1
ɧɚɡɵɜɚɸɬ ɩɨɞɯɨɞɹɳɢɦɢ ɞɪɨɛɹɦɢ ɰɟɩɧɨɣ ɞɪɨɛɢ [ɚ
a
01
1
1
a
1
a
, ɚ1, …]:
0
...,
2
= [ ɚ0, ɚ1, …, ɚ
Į
k
] (k = 1, 2, …) .
k-1
ɍɫɥɨɜɢɦɫɹ ɧɚɡɵɜɚɬɶ ɰɟɩɧɭɸ ɞɪɨɛɶ
= [ɚk, ɚ
r
k
k+1
, …]
ɨɫɬɚɬɤɨɦ ɰɟɩɧɨɣ ɞɪɨɛɢ. Ɍɨɝɞɚ ɧɟ ɫɥɨɠɧɨ ɜɢɞɟɬɶ, ɱɬɨ
, ɚ1, …] = [ɚ0, ɚ1, …, ɚ
[ɚ
0
k-1
, rk],
ɜ ɱɚɫɬɧɨɫɬɢ, ɟɫɥɢ ɚ
ɧɟ ɪɚɜɧɨ ɧɭɥɸ, ɬɨ
1
]..., , ,[
aaaa
k
a
010
r
1
11
.
0
1
],...,[
aa
k
ɐɟɩɧɵɟ ɞɪɨɛɢ ɦɨɝɭɬ ɛɵɬɶ ɤɨɧɟɱɧɵɦɢ [ɚ
, ɚ1, …]. Ʉɨɧɟɱɧɚɹ ɰɟɩɧɚɹ ɞɪɨɛɶ ɢ ɩɨɞɯɨɞɹɳɢɟ ɞɪɨɛɢ ɹɜɥɹɸɬɫɹ ɨɛɵɤɧɨɜɟɧ-
[ɚ
0
, ɚ1, …, ɚn] ɢ ɛɟɫɤɨɧɟɱɧɵɦɢ
0
ɧɵɦɢ ɞɪɨɛɹɦɢ. Ɉɛɨɡɧɚɱɢɦ
p
10
k
q
k
k
aaa
... ,1 ,0,] ..., , ,[
k
(11)
ɉɪɢ ɷɬɨɦ ɨɱɟɜɢɞɧɨ, ɚ
1. ɂɡɜɟɫɬɧɵ ɫɥɟɞɭɸɳɢɟ ɬɟɨɪɟɦɵ.
k
Ɍɟɨɪɟɦɚ 8. ȼɫɹɤɨɟ ɱɢɫɥɨ ɦɨɠɟɬ ɛɵɬɶ ɪɚɡɥɨɠɟɧɨ ɜ ɰɟɩɧɭɸ ɞɪɨɛɶ ɟɞɢɧ-
ɫɬɜɟɧɧɵɦ ɨɛɪɚɡɨɦ, ɢ ɜɫɹɤɚɹ ɤɨɧɟɱɧɚɹ ɢɥɢ ɛɟɫɤɨɧɟɱɧɚɹ ɰɟɩɧɚɹ ɞɪɨɛɶ ɢɦɟɟɬ
ɫɜɨɢɦ ɡɧɚɱɟɧɢɟɦ ɧɟɤɨɬɨɪɨɟ ɞɟɣɫɬɜɢɬɟɥɶɧɨɟ ɱɢɫɥɨ.
ɉɭɫɬɶ Į – ɩɪɨɢɡɜɨɥɶɧɨɟ ɱɢɫɥɨ. ɉɨɥɨɠɢɦ ɚ
ɱɚɫɬɶ Į. Ⱦɚɥɟɟ, ɟɫɥɢ Į ɧɟ ɰɟɥɨɟ ɱɢɫɥɨ, ɬɨ ɫɭɳɟɫɬɜɭɟɬ r
= [Į], ɝɞɟ [Į] – ɰɟɥɚɹ
0
> 1 ɬɚɤɨɟ, ɱɬɨ
1
1
a
D
.
0
r
1
41
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
ȿɫɥɢ r1 – ɧɟ ɰɟɥɨɟ, ɬɨ ɩɨɥɨɠɢɦ ɚ1= [r1] ɢ ɬɨɝɞɚ ɧɚɣɞɟɬɫɹ r2 ɬɚɤɨɟ, ɱɬɨ
1
ar
.
11
r
2
ɂ ɬɚɤ ɞɚɥɟɟ. ȿɫɥɢ r
– ɧɟ ɰɟɥɨɟ, ɬɨ ɚn = [rn] ɢ r
n
ɬɚɤɨɟ, ɱɬɨ
n+1
1
ar
.
nn
r
1 n
ɗɬɢ ɫɨɨɬɧɨɲɟɧɢɹ ɩɨɤɚɡɵɜɚɸɬ, ɱɬɨ
, r1] = [ɚ0, ɚ1, r2] = … = [ɚ0, ɚ1, …, ɚ
Į = [ɚ
0
n–1
, rn]
ɢ ɟɫɥɢ r
– ɰɟɥɨɟ, ɬɨ ɩɪɨɰɟɫɫ ɨɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɧɚ [ɚ0, ɚ1, …, ɚ
n
, ɚn] – ɤɨɧɟɱɧɨɣ
n-1
ɞɪɨɛɢ.
a
ȿɫɥɢ Į – ɰɟɥɨɟ, ɬɨ ɨɱɟɜɢɞɧɨ, ɱɬɨ Į = [ɚ
]. ɉɭɫɬɶ
0
D
b
ɉɪɢɦɟɧɹɹ ɚɥɝɨɪɢɬɦ ȿɜɤɥɢɞɚ, ɩɨɥɭɱɚɟɦ
rbqa
rqrb
221
rqrr
3321
a
ɬ.ɟ.,
b
b
ɬ.ɟ.,
r
ɬ.ɟ.,
1
q
,
111
b
r
1
1
q
,
2
r
1
r
1
r
2
1
r
2
1
q
,
3
r
2
r
3
...
r
2
n
rqrr
12
nnnn
qrr
11
nnn
ɬ.ɟ.,
r
1
n
r
1
n
r
n
1
q
q
,
n
r
1
n
r
n
.ɬ.ɟ.,
1
n
0 ,0 ,1 !! ba
.
42
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
Ɂɧɚɱɢɬ,
a
q
1
b
q
1
1
2
q
3
] ..., , ,[
qqq
121
n
1
q
n
,
q
1
n
ɝɞɟ q
, q2, …, q
1
– ɤɚɤ ɪɚɡ ɬɟ ɫɚɦɵɟ ɧɟɩɨɥɧɵɟ ɱɚɫɬɧɵɟ. Ɋɚɫɫɭɠɞɚɹ ɩɨ ɢɧɞɭɤ-
n+1
ɰɢɢ, ɩɨɥɭɱɢɦ, ɱɬɨ ɪɚɜɧɵɟ ɞɪɨɛɢ ɛɭɞɭɬ ɢɦɟɬɶ ɪɚɜɧɵɟ ɧɟɩɨɥɧɵɟ ɱɚɫɬɧɵɟ, ɬɨ
ɟɫɬɶ ɟɞɢɧɫɬɜɟɧɧɨɫɬɶ ɪɚɡɥɨɠɟɧɢɹ ɜ ɤɨɧɟɱɧɵɟ ɞɪɨɛɢ ɞɨɤɚɡɚɧɚ.
ȼ ɫɥɭɱɚɟ ɛɟɫɤɨɧɟɱɧɵɯ ɞɪɨɛɟɣ ɩɨɧɹɬɧɨ, ɱɬɨ ɟɫɥɢ Į – ɢɪɪɚɰɢɨɧɚɥɶɧɨɟ
ɱɢɫɥɨ, ɬɨ ɨɧɨ ɹɜɥɹɟɬɫɹ ɩɪɟɞɟɥɨɦ ɪɚɰɢɨɧɚɥɶɧɵɯ ɱɢɫɟɥ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɟɝɨ
ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɰɟɩɧɨɣ ɞɪɨɛɶɸ (ɛɟɫɤɨɧɟɱɧɨɣ)
ɬɚɤɠɟ ɨɞɧɨɡɧɚɱɧɨ. Ⱦɥɹ ɨɤɨɧɱɚɬɟɥɶɧɨɝɨ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɧɚɦ ɩɨɧɚɞɨɛɹɬɫɹ ɫɜɨɣɫɬɜɚ ɩɪɟɞɫɬɚɜɥɟɧɢɣ ɩɨɞɯɨɞɹɳɢɯ ɞɪɨɛɟɣ ɜ ɜɢɞɟ ɨɛɵɤɧɨɜɟɧɧɵɯ («ɨɞɧɨɷɬɚɠɧɵɯ») ɞɪɨɛɟɣ (11).
Ɍɟɨɪɟɦɚ 9 (Ɂɚɤɨɧ ɨɛɪɚɡɨɜɚɧɢɹ ɩɨɞɯɨɞɹɳɢɯ ɞɪɨɛɟɣ). Ⱦɥɹ ɥɸɛɨɝɨ k > 1
ɜ ɜɵɪɚɠɟɧɢɢ (11) ɢɦɟɟɦ
qqaq
ppap
kkkk
(12)
.
2121kkkk
®
¯
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. ɉɪɢ k = 2 ɮɨɪɦɭɥɵ (12) ɥɟɝɤɨ ɩɪɨɜɟɪɹɸɬɫɹ. ɉɪɟɞɩɨ-
ɥɨɠɢɦ, ɱɬɨ ɨɧɢ ɜɟɪɧɵ ɞɥɹ ɜɫɟɯ k < n. Ɋɚɫɫɦɨɬɪɢɦ ɰɟɩɧɭɸ ɞɪɨɛɶ
ɤɚɤ
211
1
a ],...,,[
0
r
1
c
aaar
],...,,[
, ɟɟ j-ɟ ɩɨɞɯɨɞɹɳɢɟ ɞɪɨɛɢ ɨɛɨɡɧɚɱɢɦ ɱɟɪɟɡ
n
d
p
n
aaa
10
, ɬɨ
n
q
n
c
j
. Ɍɨɝɞɚ, ɬɚɤ
d
j
®
cq
¯
dcap
110
nnn
1
nn
.
Ⱥ ɬɚɤ ɤɚɤ ɩɨ ɩɪɟɞɩɨɥɨɠɟɧɢɸ ɢɧɞɭɤɰɢɢ
n–2
ccac
ddad
+ d
nnnn
321
nnnn
n–2
,
321
) + (a0c
n–3
+ q
n–3
) = anp
n–1
+ p
ɬɨ
ɪn = a0(anc
n–2
+ c
) + (and
n–3
n–2
+ q
®
¯
) = an(a0c
n–3
n–2
43
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
qn = anc
n–2
+ c
n–3
= anq
n–1
+ q
n–2
.
Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ.
Ɂɚɦɟɱɚɧɢɟ. ɑɚɫɬɨ ɜɜɨɞɹɬ ɜ ɪɚɫɫɦɨɬɪɟɧɢɟ ɞɪɨɛɢ ɩɨɪɹɞɤɚ (–1), ɩɨɥɚɝɚɹ
= 1 ɢ q–1 = 0. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɜ ɷɬɨɦ ɢ ɬɨɥɶɤɨ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɮɨɪɦɭɥɵ (12)
ɪ
–1
ɫɨɯɪɚɧɹɸɬɫɹ.
ɉɭɫɬɶ
p
q
p
k
k
ɢ
– ɞɜɟ ɩɨɞɯɨɞɹɳɢɟ ɞɪɨɛɢ ɰɟɩɧɨɣ ɞɪɨɛɢ [ɚ0, ɚ1, …]
q
11k
k
ɫ ɰɟɥɵɦɢ ɧɟɩɨɥɧɵɦɢ ɱɚɫɬɧɵɦɢ. Ɍɨɝɞɚ ɞɪɨɛɶ
c
p
c
q
ppj
kk
21
qqj
kk
21
aj
..., ,1 ,0,
k
ɧɚɡɵɜɚɟɬɫɹ ɩɪɨɦɟɠɭɬɨɱɧɨɣ ɞɥɹ
c
p
p
2
k
ɬɟɨɪɟɦɵ 9, ɩɪɢ j = 0,
c
q
, ɚ ɩɪɢ j = ɚk,
q
2
k
p
q
p
1
k
k
k
ɢ
ɞɪɨɛɶɸ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ, ɜ ɫɢɥɭ
q
1
k
c
p
p
1
k
.
c
q
q
1
k
ȼɵɜɟɞɟɦ ɟɳɟ ɧɟɫɤɨɥɶɤɨ ɫɜɨɣɫɬɜ ɰɟɩɧɵɯ ɞɪɨɛɟɣ.
ɋɜɨɣɫɬɜɨ 1. Ⱦɥɹ ɜɫɟɯ k > –1
– pkq
q
kpk–1
= (–1)k.
k–1
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. ɍɦɧɨɠɚɹ ɮɨɪɦɭɥɵ (12) ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɧɚ q
k–1
ɢ p
k–1
ɢ ɜɵɱɢɬɚɹ ɩɟɪɜɭɸ ɢɡ ɜɬɨɪɨɣ, ɧɚɯɨɞɢɦ
q
kpk–1
– pkq
k–1
= –( q
k–1pk–2
– p
k–1qk–2
),
ɚ ɬɚɤ ɤɚɤ q
– p0q–1 = 1, ɬɨ ɫɜɨɣɫɬɜɨ ɞɨɤɚɡɚɧɨ.
0p–1
ɋɜɨɣɫɬɜɨ 1 ɩɨɡɜɨɥɹɟɬ ɧɚɯɨɞɢɬɶ ɪɟɲɟɧɢɟ ɞɢɨɮɚɬɨɜɚ ɭɪɚɜɧɟɧɢɹ
ax + by = 1,
ɝɞɟ a ɢ b – ɜɡɚɢɦɧɨ ɩɪɨɫɬɵɟ ɱɢɫɥɚ ɢ ɬɟɦ ɫɚɦɵɦ – ɞɥɹ ɥɸɛɨɝɨ ɫ ɧɚɯɨɞɢɬɶ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɞɢɨɮɚɬɨɜɚ ɭɪɚɜɧɟɧɢɹ: ax + by = ɫ:
®
¯
bkcxx
1
1
k
akcyy
..., ,2 ,1 ,0,
rr
ɝɞɟ aɯ1 + bɭ1 = 1.
44
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
ɋɜɨɣɫɬɜɨ 2. Ⱦɥɹ k > 0
p
p
1
k
q
k
k
)1(
k
k
.
qqq
11
kk
ɋɜɨɣɫɬɜɨ 3. Ⱦɥɹ k > 0
– pkq
q
kpk–2
k–2
= (–1)
k-1
ɚk.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. ɍɦɧɨɠɚɹ ɮɨɪɦɭɥɵ (12) ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɧɚ q
ɢ p
k-2
k-2
ɢ ɜɵɱɢɬɚɹ ɩɟɪɜɭɸ ɢɡ ɜɬɨɪɨɣ, ɧɚɯɨɞɢɦ ɜ ɫɢɥɭ ɫɜɨɣɫɬɜɚ 1:
q
kpk–2
– pkq
= ɚk( q
k–2
k–1pk–2
– p
k–1qk–2
) = (–1)
k-1
ɚk.
ɋɜɨɣɫɬɜɨ ɞɨɤɚɡɚɧɨ.
ɋɜɨɣɫɬɜɨ 4. Ⱦɥɹ k > 1
p
p
2
k
q
q
k
k
)1(
a
k
k
k
.
qq
12
kk
ɋɜɨɣɫɬɜɨ 5. ɉɨɞɯɨɞɹɳɢɟ ɞɪɨɛɢ ɱɟɬɧɨɝɨ ɩɨɪɹɞɤɚ ɨɛɪɚɡɭɸɬ ɜɨɡɪɚɫɬɚɸ-
ɳɭɸ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ, ɚ ɩɨɞɯɨɞɹɳɢɟ ɞɪɨɛɢ ɧɟɱɟɬɧɨɝɨ ɩɨɪɹɞɤɚ – ɭɛɵɜɚɸɳɭɸ. ɉɪɢ ɷɬɨɦ ɥɸɛɚɹ ɩɨɞɯɨɞɹɳɚɹ ɞɪɨɛɶ ɧɟɱɟɬɧɨɝɨ ɩɨɪɹɞɤɚ ɛɨɥɶɲɟ ɥɸɛɨɣ
ɩɨɞɯɨɞɹɳɟɣ ɞɪɨɛɢ ɱɟɬɧɨɝɨ ɩɨɪɹɞɤɚ.
ɂɡ ɷɬɨɝɨ ɫɜɨɣɫɬɜɚ ɫɥɟɞɭɟɬ, ɱɬɨ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɩɨɞɯɨɞɹɳɢɯ ɞɪɨɛɟɣ
ɫ ɱɟɬɧɵɦɢ ɧɨɦɟɪɚɦɢ ɢ ɩɨɞɯɨɞɹɳɢɯ ɞɪɨɛɟɣ ɫ ɧɟɱɟɬɧɵɦɢ ɧɨɦɟɪɚɦɢ ɫɯɨɞɹɬɫɹ,
ɫɢɥɭ ɫɜɨɣɫɬɜɚ 2 ɨɧɢ ɫɯɨɞɹɬɫɹ ɤ ɨɞɧɨɦɭ ɱɢɫɥɭ, ɬɨ ɟɫɬɶ ɫɭɳɟɫɬɜɭɟɬ ɩɪɟɞɟɥ:
ɚ ɜ
]. ..., , ,[ lim
aaa
10 n
fo
n
ɋɜɨɣɫɬɜɨ 6. Ⱦɥɹ k > 0 ɞɪɨɛɶ
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɬɟɨɪɟɦɚ 8 ɩɨɥɧɨɫɬɶɸ ɞɨɤɚɡɚɧɚ.
p
k
ɧɟɫɨɤɪɚɬɢɦɚ.
q
k
ɋɜɨɣɫɬɜɨ 7. Ⱦɥɹ k > –2
11
kk
§
§
1
¨
q
k
¨
t
¨
¨
2
5
©
©
§
·
51
¸
¸
¹
¨
¨
©
·
·
51
¸
¸
¸
¸
2
¹
¹
ɢ ɪɚɜɟɧɫɬɜɨ ɞɨɫɬɢɝɚɟɬɫɹ ɬɨɥɶɤɨ ɩɪɢ q
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. ɇɚɦ ɭɠɟ ɢɡɚɟɫɬɧɨ, ɱɬɨ q
ɱɢɫɥɚ ɢ q
= anq
n
ɥɟɣ ɛɭɞɟɬ ɧɚɛɥɸɞɚɬɶɫɹ ɩɪɢ q
ɱɚɟɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ: q
n–1
+ q
n–2
q
n–1
–1
+ q
n–2
= q
n
n–1
= 0, q0 = 1, qn = q
. ɇɚɢɛɨɥɟɟ ɦɟɞɥɟɧɧɵɣ ɪɨɫɬ ɷɬɢɯ ɡɧɚɦɟɧɚɬɟ-
+ q
= q1 = … = qk = 1.
0
= 0, q0 = 1, an – ɧɚɬɭɪɚɥɶɧɵɟ
–1
, ɬɨ ɟɫɬɶ ɩɪɢ ɚ0 = ɚ1 = … = ɚn =1. ɉɨɥɭ-
n–2
n–1
+ q
(n > 0), ɬɨ ɟɫɬɶ ɩɨɫɥɟɞɨɜɚ-
n–2
ɬɟɥɶɧɨɫɬɶ Ɏɢɛɨɧɚɱɱɢ: 1, 1, 2, 3, … . ɗɬɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɹɜɥɹɟɬɫɹ ɪɟɤɭɪ-
45
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
ɪɟɧɬɧɵɦ ɥɢɧɟɣɧɵɦ ɨɞɧɨɪɨɞɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɫ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ: q–1 = 0,
= 1. ȿɝɨ ɪɟɲɟɧɢɟ ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɜɢɞɟ: qn = ɯ
q
0
n
n–1
= ɯ
ɯ
n–2
+ ɯ
ɢɥɢ ɯ2 = ɯ + 1 ɢɥɢ ɯ2 – ɯ – 1 = 0
n+1
. Ɍɨɝɞɚ ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ:
ɉɨɫɥɟɞɧɟɟ ɭɪɚɜɧɟɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɢɦ ɭɪɚɜɧɟɧɢɟɦ ɪɟ-
51
r
ɤɭɪɪɟɧɬɧɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɟɝɨ ɪɟɲɟɧɢɹ
ɧɢɟ ɪɟɤɭɪɪɟɧɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ: q
= Ⱥ·(ɯ1)
n
2,1
n+1
x
+ B·(ɯ2)
ɨɩɪɟɞɟɥɹɟɬ ɨɛɳɟɟ ɪɟɲɟ-
2
n+1
, ɝɞɟ Ⱥ ɢ ȼ – ɤɨɧɫɬɚɧɬɵ,
ɤɨɬɨɪɵɟ ɧɚɯɨɞɢɦ ɢɡ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ:
1
BA
®
¯
0
BxAx
BA
®
¯
xxA
2121
A
°
1)(1
5
®
°
¯
.
BA
11
nn
·
·
51
¸
¸
. ɋɜɨɣɫɬɜɨ
¸
¸
2
¹
¹
Ɉɤɨɧɱɚɬɟɥɶɧɨ ɢɦɟɟɦ
§
1
¨
q
n
¨
5
©
·
§
¨
¨
2
©
§
51
¸
¨
¸
¨
¹
©
ɞɨɤɚɡɚɧɨ.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɞɪɨɛɶ [0, 1, 1, 1, …] ɫɯɨɞɢɬɫɹ ɤ ɱɢɫɥɭ, ɢɡɜɟɫɬɧɨɦɭ ɤɚɤ ɡɨ-
ɥɨɬɨɟ ɫɟɱɟɧɢɟ. ȼ ɫɚɦɨɦ ɞɟɥɟ ɨɛɨɡɧɚɱɢɦ ɟɝɨ ɱɟɪɟɡ ɭ, ɬɨɝɞɚ
151
1
x
1
1...] ,1 ,1 ,1[
2
xy
01
51
,
yxxx
2
2
x
.
681,0
|
ɋɜɨɣɫɬɜɨ 8. Ⱦɥɹ k > 0
prp
21
kkk
aa
,...],[
10
.
qrq
21
kkk
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. Ɍɚɤ ɤɚɤ [a
, a1, …] = [a0, a1, …, a
0
(ɡɚɤɨɧ ɨɛɪɚɡɨɜɚɧɢɹ ɰɟɩɧɵɯ ɞɪɨɛɟɣ) ɱɢɫɥɚ a
, ɦɨɝɭɬ ɛɵɬɶ ɥɸɛɵɦɢ ɩɨɥɨɠɢ-
j
, rk], ɚ ɜ ɬɟɨɪɟɦɟ 2
k–1
ɬɟɥɶɧɵɦɢ. Ɉɬɫɸɞɚ ɢ ɫɥɟɞɭɟɬ ɫɜɨɣɫɬɜɨ 8.
ɋɜɨɣɫɬɜɨ 9. Ⱦɥɹ k > 0
q
k
q
1
k
]..., , ,[
aaa
11
kk
.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. Ⱦɥɹ k = 1 ɷɬɨ ɫɨɨɬɧɨɲɟɧɢɟ ɨɱɟɜɢɞɧɨ, ɬɚɤ ɤɚɤ q
= 1. ɉɭɫɬɶ k > 1, ɢ ɩɭɫɬɶ ɭɠɟ ɞɨɤɚɡɚɧɨ, ɱɬɨ
ɚ q
0
= a1,
1
46
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
,
q
1
k
q
2
k
kk
] ..., , ,[
aaa
121
ɬɨɝɞɚ, ɩɨ ɬɟɨɪɟɦɟ 2,
= akq
+ q
q
k
k–1
k–2
.
Ɉɬɫɸɞɚ ɢɦɟɟɦ:
ª
q
k
q
k
1
q
k
2
a
k
q
a
«
k
1
¬
º
q
1
k
k
»
q
2
k
¼
kk
]...., ,,[,
aaa
11
ɋɜɨɣɫɬɜɨ 9 ɞɨɤɚɡɚɧɨ.
ɇɚɫɬɚɥɨ ɜɪɟɦɹ ɩɟɪɟɣɬɢ ɤ ɩɪɢɦɟɪɚɦ ɢɡɨɛɪɚɠɟɧɢɹ ɱɢɫɟɥ ɰɟɩɧɵɦɢ ɞɪɨ-
ɛɹɦɢ. ɇɚɦ ɩɨɦɨɠɟɬ ɜ ɷɬɨɦ ɫɥɟɞɭɸɳɢɣ ɚɥɝɨɪɢɬɦ:
Ⱥɥɝɨɪɢɬɦ 18 (ɪɚɡɥɨɠɟɧɢɟ ɜ ɰɟɩɧɭɸ ɞɪɨɛɶ).
z=eval(prompt('ȼɜɟɫɬɢ z','135/71')); eps=1e-12
s=''; if(z<0){z=-z; s='-'}y=floor(z); t='['+y; z-=y; c=y; x=z;
if(x<eps)t+=']'; else{p=0; q=1; p0=1; q0=0;
while(abs(z-p/q)>eps){x=1/x; y=floor(x); x-=y;
p1=p; q1=q; p=y*p+p0; q=y*q+q0; p0=p1; q0=q1; t+=','+y; }
u=t.split(','); if(u[u.length-1]-1==0){
u[u.length-2]++; t=u[0]; for(i=1; i<u.length-1; i++)t+=', '+u[i]; }}
s+t+']='+s+c+'.'+p+'/'+q+'='+s+(c+z)+ ', eps='+eps
Ɂɞɟɫɶ ɜɜɨɞɢɬɫɹ ɱɢɫɥɨ ɯ ɢɥɢ ɟɝɨ ɚɥɝɟɛɪɚɢɱɟɫɤɨɟ ɜɵɪɚɠɟɧɢɟ, ɧɚ ɜɵɯɨɞɟ
ɩɨɥɭɱɚɟɬɫɹ ɢɡɨɛɪɚɠɟɧɢɟ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɯ ɫ ɬɨɱɧɨɫɬɶɸ eps = 10
ɞɪɨɛɶɸ [a
, a1, …, a
[a
0
, a1, … a
0
k–1
, ak], ɩɪɢ ɷɬɨɦ, ɟɫɥɢ ak = 1, ɬɨ ɚɥɝɨɪɢɬɦ ɜɨɡɜɪɚɳɚɟɬ ɞɪɨɛɶ
k–1
+ 1]. Ɍɨɱɧɨɫɬɶ eps ɦɨɠɧɨ ɦɟɧɹɬɶ, ɜɤɥɸɱɚɹ ɟɟ ɜ ɚɥɝɨɪɢɬɦ. ȼɜɨɞ
–12
, ɰɟɩɧɨɣ
ɷɥɟɦɟɧɬɚɪɧɵɯ ɮɭɧɤɰɢɣ ɩɪɨɢɡɜɨɞɢɬɫɹ ɭɱɢɬɵɜɚɹ ɫɢɧɬɚɤɫɢɫ JavaScript. ɇɚɩɪɢ-
S
sin
ɦɟɪ, ɜɵɱɢɫɥɹɹ
ɦɵ ɜɜɟɞɟɦ: "Math.sin(Math.PI/6)". ɇɨ ɧɢɝɞɟ ɜ ɚɥɝɨɪɢɬɦɟ
6
ɧɟ ɩɢɲɢɬɟ ɫɥɨɜɨ "Math.", ɚ ɬɨɥɶɤɨ ɩɪɢ ɜɜɨɞɟ.
Ɍɟɩɟɪɶ ɜɜɟɞɟɦ ɜ ɧɚɲɢ ɢɫɫɥɟɞɨɜɚɧɢɹ ɟɳɟ ɨɞɢɧ ɨɛɴɟɤɬ, ɧɚɡɵɜɚɟɦɵɣ ɤɨɧ-
ɬɢɧɭɚɧɬɨɣ. Ɉɧ ɩɨɧɚɞɨɛɢɬɫɹ ɧɚɦ ɞɥɹ ɨɰɟɧɤɢ ɫɥɨɠɧɨɫɬɢ ɚɥɝɨɪɢɬɦɚ ȿɜɤɥɢɞɚ.
Ʉɨɧɬɢɧɭɚɧɬɚ (ɚ
ɝɨɧɚɥɶɸ diag(ɚ
0
, ɚ1, …, ɚn) – ɷɬɨ ɨɩɪɟɞɟɥɢɬɟɥɶ ɦɚɬɪɢɰɵ ɫ ɝɥɚɜɧɨɣ ɞɢɚ-
0
, ɚ1, …, ɚn), ɜɟɪɯɧɟɣ ɧɚɞɞɢɚɝɨɧɚɥɶɸ, ɫɨɫɬɨɹɳɟɣ ɢɡ ɟɞɢɧɢɰ
ɢ ɩɨɞɞɢɚɝɨɧɚɥɶɸ – ɢɡ (–1):
47
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
a
0
a
1
a
aaa
),...,,(
n
10
2
...
0...001
0...0011
0...010
a
11...0
n
1
a
10...0
n
Ɋɚɫɤɥɚɞɵɜɚɹ ɷɬɨɬ ɨɩɪɟɞɟɥɢɬɟɥɶ ɩɨ ɩɨɫɥɟɞɧɟɦɭ ɫɬɨɥɛɰɭ, ɩɨɥɭɱɢɦ
(ɚ0, ɚ1, …, ɚn) = ɚn(ɚ0, …, ɚ
) + (ɚ0, …, ɚ
n-1
). (13)
n-2
Ɉɬɫɸɞɚ ɦɟɬɨɞɨɦ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɢɧɞɭɤɰɢɢ ɢɦɟɟɦ ɥɟɦɦɭ.
Ʌɟɦɦɚ 1. Ʉɨɧɬɢɧɭɚɧɬɚ (ɚ
ɜɟɞɟɧɢɣ ɷɥɟɦɟɧɬɨɜ ɚ
, ɚ1, …, ɚn, ɨɞɧɨ ɢɡ ɤɨɬɨɪɵɯ ɫɨɞɟɪɠɢɬ ɜɫɟ ɷɬɢ ɷɥɟɦɟɧɬɵ,
0
, ɚ1, …, ɚn) ɪɚɜɧɚ ɫɭɦɦɟ ɜɫɟɜɨɡɦɨɠɧɵɯ ɩɪɨɢɡ-
0
ɚ ɞɪɭɝɢɟ ɩɨɥɭɱɚɸɬɫɹ ɢɡ ɧɟɝɨ ɜɵɛɪɚɫɵɜɚɧɢɟɦ ɨɞɧɨɣ ɢɥɢ ɧɟɫɤɨɥɶɤɢɯ ɩɚɪ ɫɨɦɧɨɠɢɬɟɥɟɣ ɫ ɫɨɫɟɞɧɢɦɢ ɧɨɦɟɪɚɦɢ (ɟɫɥɢ ɜɵɛɪɨɫɢɥɢ ɜɫɟ ɫɨɦɧɨɠɢɬɟɥɢ, ɬɨ
ɫɱɢɬɚɟɦ, ɱɬɨ ɨɫɬɚɥɚɫɶ ɟɞɢɧɢɰɚ).
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. Ⱦɥɹ k = 1 ɥɟɦɦɚ ɜɟɪɧɚ:
1
a
0
aa
),(
aa
10
1
a
1
1
.
10
ɉɪɟɞɩɨɥɨɠɢɦ, ɥɟɦɦɚ ɜɟɪɧɚ ɞɥɹ ɜɫɟɯ k < n, ɬɨɝɞɚ ɢɡ ɪɚɜɟɧɫɬɜɚ (13) ɜɵɬɟɤɚɟɬ ɭɬɜɟɪɠɞɟɧɢɟ ɥɟɦɦɵ ɞɥɹ k = n.
ɋɥɟɞɫɬɜɢɟ. Ʉɨɥɢɱɟɫɬɜɨ ɫɥɚɝɚɟɦɵɯ ɜ ɤɨɧɬɢɧɭɚɧɬɟ n-ɝɨ ɩɨɪɹɞɤɚ ɟɫɬɶ
ɫɭɦɦɚ ɱɢɫɥɚ ɫɥɚɝɚɟɦɵɯ ɜ ɤɨɧɬɢɧɭɚɧɬɚɯ (n-1)-ɝɨ ɢ (n-2)-ɝɨ ɩɨɪɹɞɤɨɜ, ɬɨ ɟɫɬɶ
ɤɨɧɬɢɧɭɚɧɬɚ (ɚ
, ɚ1, …, ɚ
0
) ɫɨɞɟɪɠɢɬ F
n-1
ɫɥɚɝɚɟɦɵɯ, ɝɞɟ Fn – (n+1)-ɟ ɱɢɫɥɨ
n+1
Ɏɢɛɨɧɚɱɱɢ.
),...,,(
aaa
10
Ʌɟɦɦɚ 2 (Ʌ. ɗɣɥɟɪ).
],...,,[
aaa
10
n
n
),...,,(
aaa
21
n
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. Ⱦɥɹ k = 1ɥɟɦɦɚ ɜɟɪɧɚ:
),(1
],[
1
aaa
010
a
1
aa
10
a
1
aa
10
.
)(
a
1
ɉɪɟɞɩɨɥɨɠɢɦ, ɥɟɦɦɚ ɜɟɪɧɚ ɞɥɹ k = n – 1, ɬɨɝɞɚ
),...,,(
aaa
110
n
],...,,[
aaa
110
n
.
),...,,(
aaa
121
n
48
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
Ɉɬɫɸɞɚ ɢ (13) ɢɦɟɟɦ (ɚ0, …, ɚ
n–1
)
§
¨
¨
ª
«
¬
§
¨
a
n
¨
©
§
¨
a
n
¨
©
0
·
1
¸
¸
a
n
¹
·
1
¸
¸
a
n
¹
aa
),...,(
n
.
aa
),...,(
11
n
§
¨
aaaaaa
,...,,],...,,[
11010
nn
¨
©
º
·
1
©
¸
»
¸
a
§
n
¹
¼
¨
¨
©
aaaa
),...,(),...,(
30201
nn
aaaa
),...,(),...,(
31211
nn
§
¨
aaa
,...,,
,...,,
110
n
¨
©
§
¨
aaa
121
n
¨
©
aa
),...,(
n
aa
),...,(
n
·
·
1
¸
¸
¸
¸
a
n
¹
¹
·
·
1
¸
¸
¸
¸
a
n
¹
¹
1
a
n
1
a
n
aa
),...,(
2010
n
aa
),...,(
2111
n
Ʌɟɦɦɚ 2 ɞɨɤɚɡɚɧɚ.
ɍɬɜɟɪɠɞɟɧɢɟ ɥɟɦɦɵ ɭɫɬɚɧɚɜɥɢɜɚɟɬ ɫɜɹɡɶ ɦɟɠɞɭ ɰɟɩɧɵɦɢ ɞɪɨɛɹɦɢ
ɢ ɤɨɧɬɢɧɭɚɧɬɚɦɢ ɢ ɩɪɟɞɫɬɚɜɥɹɟɬ ɰɟɩɧɭɸ ɞɪɨɛɶ ɜ ɜɢɞɟ ɨɛɵɤɧɨɜɟɧɧɨɣ ɞɪɨɛɢ.
ɉɪɢɫɬɭɩɢɦ ɬɟɩɟɪɶ ɤ ɨɰɟɧɤɟ ɫɥɨɠɧɨɫɬɢ ɚɥɝɨɪɢɬɦɚ ȿɜɤɥɢɞɚ. ɉɨɞ «ɞɨɪɨɝɨɣ» ɨɩɟɪɚɰɢɟɣ ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɨɩɟɪɚɰɢɸ ɞɟɥɟɧɢɹ ɫ ɨɫɬɚɬɤɨɦ (ɮɭɧɤɰɢɹ x%y –
ɜ ɚɥɝɨɪɢɬɦɟ 11). Ȼɭɞɟɦ ɢɫɤɚɬɶ ɧɚɢɯɭɞɲɢɣ ɫɥɭɱɚɣ – ɤɨɝɞɚ ɚɥɝɨɪɢɬɦ ɪɚɛɨɬɚɟɬ
ɨɫɨɛɟɧɧɨ ɞɨɥɝɨ. Ⱦɨɤɚɠɟɦ ɫɥɟɞɭɸɳɭɸ
ɬɟɨɪɟɦɭ.
Ɍɟɨɪɟɦɚ 10 (ɬɟɨɪɟɦɚ Ʌɚɦɷ, 1845 ɝ.). ɉɭɫɬɶ nN ɢ a>b>0 ɬɚɤɢɟ, ɱɬɨ ɚɥɝɨɪɢɬɦɭ ȿɜɤɥɢɞɚ ɧɟɨɛɯɨɞɢɦɨ ɜɵɩɨɥɧɢɬɶ ɬɨɱɧɨ n ɞɟɥɟɧɢɣ ɫ ɨɫɬɚɬɤɨɦ, ɩɪɢɱɟɦ
ɚ – ɧɚɢɦɟɧɶɲɟɟ ɫ ɬɚɤɢɦ ɫɜɨɣɫɬɜɨɦ. Ɍɨɝɞɚ ɚ=F
n+2
, b= F
, ɝɞɟ Fk – ɱɢɫɥɚ Ɏɢ-
n+1
ɛɨɧɚɱɱɢ.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. Ɋɚɡɥɨɠɢɦ ɱɢɫɥɨ
a
aaa
b
ɇɚɩɨɦɧɢɦ, ɱɬɨ ɚ0, ɚ1, …, ɚ
n–1
a
ɜ ɰɟɩɧɭɸ ɞɪɨɛɶ
b
) ..., , ,(
aaa
110
n
] ..., , ,[
110
n
.
) ..., , ,(
aaa
121
n
– ɧɟɩɨɥɧɵɟ ɱɚɫɬɧɵɟ ɢɡ ɚɥɝɨɪɢɬɦɚ ȿɜɤɥɢɞɚ
ɢ ɢɯ n ɲɬɭɤ.
, …, ɚ
(ɚ
0
ɮɢɰɢɟɧɬɚɦɢ ɨɬ ɷɬɢɯ ɩɟɪɟɦɟɧɧɵɯ ɚ
ɢ ɩɨɫɥɟɞɧɢɣ ɚ
= ɚ1 = … = ɚ
ɢ ɚ
0
ɢ b = F
n+1
) – ɤɨɧɬɢɧɭɚɧɬ, ɬɨ ɟɫɬɶ ɦɧɨɝɨɱɥɟɧ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵɦɢ ɤɨɷɮ-
n–1
1. ɉɨɷɬɨɦɭ ɟɝɨ ɦɢɧɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɛɭɞɟɬ ɩɪɢ ɚ
n–1
= 1. Ɉɬɫɸɞɚ ɢ ɫɥɟɞɫɬɜɢɹ ɤ ɥɟɦɦɟ 1 ɩɨɥɭɱɚɟɦ, ɱɬɨ a = F
n–2
, …, ɚ
0
, ɩɪɢɱɟɦ ɚ0 > 0, ɬɚɤ ɤɚɤ a > b
n–1
n–1
= 2
n+2
(ɬɨ, ɱɬɨ ɤɨɧɬɢɧɭɚɧɬɵ (1, 1, …, 1, 2) ɟɫɬɶ ɱɢɫɥɚ Ɏɢɛɨɧɚɱɱɢ, ɥɟɝɤɨ
ɞɨɤɚɡɚɬɶ ɢɧɞɭɤɰɢɟɣ ɢɡ ɮɨɪɦɭɥɵ (13)). Ɍɟɨɪɟɦɚ Ʌɚɦɷ ɞɨɤɚɡɚɧɚ.
49
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
ɋɥɟɞɫɬɜɢɟ. ȿɫɥɢ ɧɚɬɭɪɚɥɶɧɵɟ ɱɢɫɥɚ a ɢ b ɧɟ ɩɪɟɜɨɫɯɨɞɹɬ NN, ɬɨ ɱɢɫɥɨ ɨɩɟɪɚɰɢɣ ɞɟɥɟɧɢɹ ɫ ɨɫɬɚɬɤɨɦ, ɧɟɨɛɯɨɞɢɦɵɯ ɚɥɝɨɪɢɬɦɭ ȿɜɤɥɢɞɚ ɞɥɹ ɨɛɪɚ-
15log
N
ɛɨɬɤɢ ɦɟɧɶɲɟ, ɱɟɦ
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. ɗɬɨ ɫɥɟɞɫɬɜɢɟ ɜɵɬɟɤɚɟɬ ɢɡ ɫɜɨɣɫɬɜɚ 7 ɢ ɟɝɨ ɞɨɤɚɡɚ-
ɬɟɥɶɫɬɜɚ.
ɉɪɨɜɟɪɶɬɟ, ɱɬɨ ɫ ɱɢɫɥɚɦɢ a ɢ b ɦɟɧɶɲɟ ɦɢɥɥɢɨɧɚ ɚɥɝɨɪɢɬɦ ȿɜɤɥɢɞɚ
«ɫɩɪɚɜɢɬɫɹ» ɧɟ ɛɨɥɟɟ ɱɟɦ ɡɚ 28 ɲɚɝɨɜ.
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɨ ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɢɪɪɚɰɢɨɧɚɥɶɧɨɝɨ ɱɢɫɥɚ ɫ ɬɨɣ ɢɥɢ
ɢɧɨɣ ɫɬɟɩɟɧɶɸ ɬɨɱɧɨɫɬɢ ɜ ɜɢɞɟ ɨɛɵɤɧɨɜɟɧɧɨɣ ɞɪɨɛɢ, ɤɨɬɨɪɚɹ ɢɦɟɟɬ ɝɥɭɛɨɤɢɟ
ɤɨɪɧɢ. ɇɚɩɪɢɦɟɪ, ɪɟɡɭɥɶɬɚɬɵ ɜɵɱɢɫɥɟɧɢɣ ɤɨɦɩɶɸɬɟɪɚ
ɫɹ ɞɟɫɹɬɢɱɧɵɦɢ ɞɪɨɛɹɦɢ ɫ ɧɟɤɨɬɨɪɨɣ ɬɨɱɧɨɫɬɶɸ, ɯɨɬɹ ɬɨɱɧɵɣ ɪɟɡɭɥɶɬɚɬ ɦɨɠɟɬ ɛɵɬɶ ɪɚɰɢɨɧɚɥɶɧɵɦ ɱɢɫɥɨɦ. ɂɥɢ ɦɵ ɯɨɬɢɦ ɡɚɦɟɧɢɬɶ ɢɪɪɚɰɢɨɧɚɥɶɧɨɟ
ɱɢɫɥɨ ɧɚ ɟɝɨ ɪɚɰɢɨɧɚɥɶɧɨɟ ɩɪɢɛɥɢɠɟɧɢɟ, ɤɚɤ, ɧɚɩɪɢɦɟɪ, Ⱥɪɯɢɦɟɞ ɩɪɢɛɥɢ-
ɠɟɧɧɨ ɩɪɟɞɫɬɚɜɥɹɥ
ȼ ɫɜɨɟɦ ɩɪɨɫɬɟɣɲɟɦ ɜɢɞɟ ɷɬɚ ɡɚɞɚɱɚ ɨɛɵɱɧɨ ɫɬɚɜɢɬɫɹ ɬɚɤ, ɱɬɨ ɢɳɟɬɫɹ
ɪɚɰɢɨɧɚɥɶɧɚɹ ɞɪɨɛɶ ɫ ɧɚɢɦɟɧɶɲɢɦ (ɩɨɥɨɠɢɬɟɥɶɧɵɦ) ɡɧɚɦɟɧɚɬɟɥɟɦ, ɨɬɥɢɱɚɸɳɚɹɫɹ ɨɬ ɞɚɧɧɨɝɨ ɱɢɫɥɚ ɧɟ ɛɨɥɟɟ ɱɟɦ ɧɚ ɧɟɤɨɬɨɪɭɸ ɧɚɩɟɪɟɞ ɡɚɞɚɧɧɭɸ ɜɟɥɢɱɢɧɭ.
ɇɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɹɫɧɨ, ɱɬɨ ɞɥɹ ɪɟɲɟɧɢɹ ɷɬɨɣ ɡɚɞɚɱɢ ɚɩɩɚɪɚɬ ɫɢɫɬɟɦɚɬɢɱɟɫɤɢɯ ɞɪɨɛɟɣ ɫɨɜɟɪɲɟɧɧɨ ɧɟ ɩɪɢɝɨɞɟɧ, ɢɛɨ ɞɚɜɚɟɦɵɟ ɢɦ ɚɩɩɪɨɤɫɢɦɢɪɭɸɳɢɟ
(ɜ ɫɥɭɱɚɟ ɞɟɫɹɬɢɱɧɵɯ ɞɪɨɛɟɣ ɷɬɨ ɫɬɟɩɟɧɢ ɱɢɫɥɚ 10) ɢ ɫɨɜɟɪɲɟɧɧɨ ɧɟ ɡɚɜɢɫɹɳɢɟ ɨɬ ɚɪɢɮɦɟɬɢɱɟɫɤɨɣ ɩɪɢɪɨɞɵ ɢɡɨɛɪɚɠɚɟɦɨɝɨ ɱɢɫɥɚ. ɇɚɩɪɨɬɢɜ, ɜ ɫɥɭɱɚɟ
ɰɟɩɧɵɯ ɞɪɨɛɟɣ ɡɧɚɦɟɧɚɬɟɥɢ ɩɨɞɯɨɞɹɳɢɯ ɞɪɨɛɟɣ ɩɨɥɧɨɫɬɶɸ ɨɩɪɟɞɟɥɹɸɬɫɹ
ɢɡɨɛɪɚɠɚɟɦɵɦ ɱɢɫɥɨɦ, ɢ ɩɨɷɬɨɦɭ ɦɵ ɢɦɟɟɦ ɜɫɟ ɨɫɧɨɜɚɧɢɹ ɨɠɢɞɚɬɶ, ɱɬɨ ɷɬɢ
ɩɨɞɯɨɞɹɳɢɟ ɞɪɨɛɢ ɞɨɥɠɧɵ ɢɝɪɚɬɶ ɫɭɳɟɫɬɜɟɧɧɭɸ ɪɨɥɶ
ɨ ɧɚɢɥɭɱɲɟɣ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɷɬɨɝɨ ɱɢɫɥɚ ɪɚɰɢɨɧɚɥɶɧɵɦɢ ɞɪɨɛɹɦɢ.
ɫɟɥ ɛɵɥɢ ɩɪɟɞɥɨɠɟɧɵ Ɇ. ȼ. Ɉɫɬɪɨɝɪɚɞɫɤɢɦ ɧɟɡɚɞɨɥɝɨ ɞɨ ɟɝɨ ɫɦɟɪɬɢ. ȿɝɨ
ɤɪɚɬɤɢɟ ɡɚɦɟɱɚɧɢɹ ɧɚ ɷɬɨɬ ɫɱɟɬ ɛɵɥɢ ɧɚɣɞɟɧɵ ɧɚ ɤɥɨɱɤɚɯ ɛɭɦɚɝɢ
ɜ ɪɭɤɨɩɢɫɧɨɦ ɮɨɧɞɟ Ⱥɤɚɞɟɦɢɢ ɧɚɭɤ ɍɤɪɚɢɧɫɤɨɣ ɋɋɊ. ɗɬɢ ɡɚɦɟɬɤɢ ɛɵɥɢ
ɪɚɫɲɢɮɪɨɜɚɧɵ ɜ ɫɬɚɬɶɟ
ɝɭɬ ɛɵɬɶ ɫɜɹɡɚɧɵ ɫ ɞɜɭɦɹ ɚɥɝɨɪɢɬɦɚɦɢ Ɇ. ȼ. Ɉɫɬɪɨɝɪɚɞɫɤɨɝɨ ɞɥɹ ɩɪɢɛɥɢɠɟɧɢɹ ɢɪɪɚɰɢɨɧɚɥɶɧɵɯ ɱɢɫɟɥ». Ʉɚɤ ɜɵɹɫɧɢɥ ȿ. ə. Ɋɟɦɟɡ, ɚɥɝɨɪɢɬɦɵ Ɇ. ȼ. Ɉɫɬɪɨɝɪɚɞɫɤɨɝɨ ɜ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɞɚɸɬ ɩɪɢɛɥɢɠɟɧɢɹ ɥɭɱɲɢɟ, ɱɟɦ ɰɟɩɧɵɟ
ɞɪɨɛɢ. Ʉ ɫɨɠɚɥɟɧɢɸ, ɞɨ ɫɢɯ ɩɨɪ ɞɟɬɚɥɶɧɨɟ ɢɯ ɢɡɭɱɟɧɢɟ, ɜ
ɜɵɱɢɫɥɢɬɟɥɶɧɵɯ ɰɟɥɟɣ, ɧɟ ɨɫɭɳɟɫɬɜɥɟɧɨ.
ɱɢɫɥɚ Į, ɟɫɥɢ ɜɫɹɤɚɹ ɞɪɭɝɚɹ ɞɪɨɛɶ ɫ ɬɟɦ ɠɟ ɢɥɢ ɦɟɧɶɲɢɦ ɡɧɚɦɟɧɚɬɟɥɟɦ ɥɟɠɢɬ
ɞɪɨɛɢ ɢɦɟɸɬ ɡɧɚɦɟɧɚɬɟɥɢ, ɨɩɪɟɞɟɥɹɟɦɵɟ ɫɢɫɬɟɦɨɣ ɫɱɢɫɥɟɧɢɹ
Ɉɞɧɚɤɨ ɞɜɚ ɢɧɬɟɪɟɫɧɵɯ ɚɥɝɨɪɢɬɦɚ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɢɪɪɚɰɢɨɧɚɥɶɧɵɯ ɱɢ-
ɇɚɡɨɜɟɦ ɪɚɰɢɨɧɚɥɶɧɨɟ ɱɢɫɥɨ ɢɥɢ ɞɪɨɛɶ
15
2
22
S
.
7
ȿ. ə. Ɋɟɦɟɡɚ «Ɉ ɡɧɚɤɨɩɟɪɟɦɟɧɧɵɯ ɪɹɞɚɯ, ɤɨɬɨɪɵɟ ɦɨ-
.
ɨɛɵɱɧɨ ɩɪɟɞɫɬɚɜɥɹɸɬ-
ɜ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ
ɬɨɦ ɱɢɫɥɟ ɢ ɞɥɹ
a
ɧɚɢɥɭɱɲɢɦ ɩɪɢɛɥɢɠɟɧɢɟɦ
b
50
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