ЛЕКЦІЇ ² ВПРАВИ

S = a + a + + a
n
1 2
K
n;
( )
a + a 2a
d n
n
1
+ −1
1
Sn
= n;
Sn
=
n.
2 2
14.2.11. Äåêàðòîâ³ êîîðäèíàòè íà ïëîùèí³
(ðèñ. 14.1 ³ 14.2):
14.2.8. Ãåîìåòðè÷íà ïðîãðåñ³ÿ
b, b , b , , b n
,
K K 1 2 3
K K ( )
n−1
b = bq ; q = b / b = = b / b = b ≠
n 1 2 1 n n−1 1
0
n n− n+
( )
b = b
1b 1
n≥
2 ; bbn = bbn
−
=
S = b1 + b2 + K + b ;
1 2 1
K;
n n
( −1) ( 1−
) ( )
n n
b1 q b1
q
Sn
= nb1
( q = 1)
; Sn
= = q ≠1
.
q−1 1−q
14.2.9. Íåñê³í÷åííî ñïàäíà ãåîìåòðè÷íà ïðîãðåñ³ÿ
( )
b, b , b , K, b , K ; q < 1 b ≠ 0 ;
1 2 3 n
1
Ðèñ. 14.1 Ðèñ. 14.2
x — àáñöèñà òî÷êè M; y — îðäèíàòà òî÷êè M; x, y —
êîîðäèíàòè òî÷êè M (ðèñ. 14.1); OM
uuuur , OM
uuuur (x, y) — âåêòîð;
x — àáñöèñà âåêòîðà OM
uuuur ; y — îðäèíàòà âåêòîðà OM
uuuur ;
x, y — êîîðäèíàòè âåêòîðà OM
uuuur (ðèñ. 14.2).
Çíàêè êîîðäèíàò òî÷êè M ³ âåêòîðà OM
uuuur çîáðàæåíî íà
ðèñ. 14.3.
b1
Sn = b1+ b2
+ K + bn; S = lim Sn;
S =
n→∞
1 − q
.
14.2.10. Á³íîì Íüþòîíà
( ) n 0 n 1 n−1 m n−m m n−1 n−1
n n
a+ b = C a + C a b+ + C a b + + C ab + C b
K K ,
n n n n n
C
m
n
( −1)( −2) K( − + 1)
n n n n m n!
An
= = =
123 ⋅ ⋅ ⋅K⋅m m! n−m ! P
( )
m
n!
An
= n( n−1)( n−2) K ( n− m+ 1)
=
!
,
P = 1⋅2⋅3 ⋅K
⋅ m = m !.
m
( n−
m)
m
m
,
Ðèñ. 14.3
14.3. ÒÐÈÃÎÍÎÌÅÒвß
14.3.1. Îçíà÷åííÿ òðèãîíîìåòðè÷íèõ ôóíêö³é
sin α= y
x
y; cos
x
OM
= α= OM
= (íà îäèíè÷íîìó êîë³
OM =1);
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