さくらの個別指導 ( さくら教育研究所 ) A a 1 a 2 a 3 a n {a n } a 1 a n n n 1 n n 0 a n = 1 n 1 n n O n {a n } n a n α {a n } α {a

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1 ... A a a a 3 a n {a n } a a n n 3 n n n 0 a n = n n n O n {a n } n a n α {a n } α {a n } α α {a n } a n n a n α a n = α n n 0 n = 0 3

2 4. () (0.) n () ( 0.) n 0 0 c c c c c c c. () n 3 n () ( )n 3 n (3) cos π cos 3π cos 5π cos(n )π B {a n } {a n } 3 a n 4 8 n a n = n 8 n n 4 {a n } {a n } O 3 n n a n a n =

3 n n 3n O 3 3 a n n {a n } 6 {a n } 9 a n = 3n n a n a n = n =, ( 3n) = 3 ( ) n n ( ) n a n = ( ) n 3 O 4 n 3 a n α α. n () n () n (3) n (4) + ( ) n

4 6 C () {a n } {b n } () a n = α b n = β k a n = k α k (a n + b n ) = α + β (a n b n ) = α β 3 a n b n = αβ a n 4 = α b n β β 0. a n = b n = 3 (3a n + b n ) = 3 a n + b n = 3 + ( 3) = 3 (a n ) = a n = = 0 b n + 4 (b n + 4) a n 3 = (a n 3) = =.3 a n = b n = () (a n b n ) () (3a n + b n ) (3) (a n ) (4) a n b n (5) b n + 5 a n (6) a n b n a n + b n } {a n } {b n } {a n + b n } {a n b n } {a n b n } { an b n

5 .. 7 {a n } {b n } a n = b n = (a n + b n ) =, a n b n = (a n b n ) a n b n n.3 () n + = + n = n () 3n n = 3 n ( (3) (n 3n) = n 3 ) n = 0 n n =.4 () n + 3n n = 3 «= n () 4n n + 3 (3) (n 3 n )

6 8. ( n + n n) ( ( n + n n)( n n + n n) = + n + n) n + n + n (n + n) n = n + n + n = = + n = + n n + n + n.5 () ( n + n) () ( n n n)

7 .. 9 D () () 5 n a n b n a n = α b n = β α β 6 n a n b n a n = b n = 7 n a n c n b n a n = b n = α c n = α 5 a n < b n α = β a n = n b n = + n a n = b n = 7. n sin nπ 3 sin nπ 3 ( ) n n n sin nπ 3 n = 0 n = 0 7 n sin nπ 3 = 0.6 θ nθ () sin nθ () cos n n 6

8 30.. A {r n } r r {r n } r > r = + h h > 0 r n = ( + h) n ( + h) n = n C 0 + n C h + n C h + + n C n h n h > 0 r n + nh = + nh + n(n ) h + + h n ( + nh) = 6 rn = r = r n = n = = 3 0 < r < r = s s > rn = s n sn = rn = s n = 0 4 r = 0 r n = 0 n = 0 = 0 5 < r < 0 r = s 0 < s < r n = s n 3 sn = 0 rn = 0 r n = 0 6 r = {r n } {r n } 7 r < r = s s > r n = ( ) n s n rn = s n = {r n }

9 .. 3 {r n } {r n } r > n! r n = r = n! r n = r < n! r n = 0 r.4 () {( ) n } > ( ) n = {( () ) n } ( < n = 0 ) (3) {( ) n } <.7 n () ( ( ) n 3 ) n () 3 (3) ( 4 3) n (4) ( ) n B {r n } {r n } {r n } < r {( ) n }.5 < <

10 3.8 {( ) n }.3 4 n + 3 n 4 n () () 4 n 3 n 3 n + n ( ) n 3 4 n + 3 n + () 4 n 3 = n 4 ( 3 4 ( 4 4 n () 3 n + = 3 n + ) n ( 3 ) n = 4 n ) n = 3 n.9 5 n n () 5 n + n 4 n n () 3 n n+ (3) 3 n n

11 .. 33 { } r n. + r n () r < () r < r n () + r = ( ) n n + r () r < rn = 0 r n + r = 0 n + 0 = 0 () r < r < r n + r n = ( r ) n + ( ) n = 0 r = 0 + = { } r n.0 + r n () r > () r = (3) r < (4) r < C {a n } a n+ = pa n + q a

12 34. {a n } a =, a n+ = a n + (n =,, 3,...) α α = α + α (a n α) = 0 α = α + a n+ = (a n ) a n+ α = (a n α) {a n } a = = ( ) n a n = ( ) ( ) n = 0 (a n ) = 0 a n = p a n+ = pa n + q α = pα + q α a n+ α = p(a n α). {a n } a =, a n+ = 3 a n + (n =,, 3,...)

13 A a a a 3 a n a + a + a a n + a n n= a a n n {a n } n S n S = a S = a + a S 3 = a + a + a 3 S n = a + a + a a n S n n S n n {S n } S S S 3 S n {S n } S S n = n a k = S k= S S S a n n= a + a + a a n + S n {S n } S S {S n }

14 n(n + ) + n(n + ) n(n + ) = n n + n S n S n = n(n + ) ( = ) ( + ) ( ) = n + S n = ( ) = n + ( n ) n n(n + ) +

15 .. 37 B a r a + ar + ar + + ar n + a r a r a + ar + ar + + ar n + n S n a = 0 {S n } 0 a 0 r = S n = a + a + a + + a = na a 0 {S n } r S n = a + ar + ar + + ar n = a( rn ) r = a r a r rn r < rn = 0 S n = a r r r > {r n } {S n } a + ar + ar + + ar n + a 0 a r < r r a = 0 0

16 38.5 () () 3 () 3 < = = () >.3 () () (3) (4) ( + ) + + ( ) +

17 ( ) + ( ) + = 0 < < < < 0 < < 0 <.4 () + ( ) + ( ) + () + ( ) + ( ) +

18 40 C.3 P O 3 P P,, +, + 3, P < O ( ) = 3 P 3.5 P O 4 6 P

19 .. 4 D.4 a ABC A B C A B C A B C C A C 3 B 3 A B C A B C A 3 B 3 C 3 A n B n C n B A 3 C S B A A B C : : A n B n C n S n S = 4 a, S = 4 S, S 3 = 4 S, S + S + S S n + 4 a 4 4 < S S = 4 a = 3 a ABC b A B C A B C A 3 B 3 C 3 A n B n C n l

20 4 E a n b n 6 n= n= () a n = S b n = T n= n= ka n = ks n= n= k (a n + b n ) = S + T (a n b n ) = S T (.7 ) n 3 n n= n= n n= 3 n 3 3 n= < 3 < n= n = =, n= 3 n = 3 3 = n= ( ) = n 3 n =.7 ( () + ) n 3 n () n 3 n 4 n n= n=

21 () (n 4n 3 ) () n n n + (3) 3 n + n+ n 3 n n + + n = () () (3) 0 [ = n + n S n = ] n + n + + n 3 7 [ = = ]

22 44... () A f() = 3 y y = 3 f() f() a a O 3 f() α α a f() a f() α a f() = α () f() = α g() = β a a a k f() = k α k a {f() + g()} = α + β a {f() g()} = α β 3 a {f()g()} = αβ 4 a f() g() = α β β 0 a f() f() = f(a) a

23 () ( + 3 4) = = 6 () = ( ) + = + 3 (3) = 4 =.8 () ( 3 ) () ( 3)( + ) (3) (4) 58 B f() = a a.7 f() = = y y = f() f() = ( + )( ) = + f() O

24 46.8 ( 3 + () () + 0 ) () + = ( + )( + ) + = ( + ) = 3 ( () 0 ) = ( + ) = 0 ( + ) = 0 ( + ) = 4.9 (3) a 0 9 () () ( (3) 0 a ) a +

25 ( + 4) = 0 ( ) = 0 ( ) = = () () 4 4

26 48.5 a b a + b = ( ) 0 ( ) 0 a + b = ( ) = 0 (a + b) = 0 a + b = 0 b = a a + b = a( ) a( )( + ) = ( )( + ) = a + = a a = a = 4 b = 4 ( ) a = 4 b = 4

27 a b a b 0 =

28 50 C f() a a f() a f() a f() a f() f() = a a a f() a f() a f() a f() f() = a.8 0 = { } = ( ) y O y =. { } () () ( ) ( + )

29 .. 5 D f() a < a > a.9 f() = + > 0 f() = + < 0 = + f() = + = y O > 0 0 f() < 0 0 f() > a a f() α α a f() β f() = α a+0 y α O a y = f() < a a β f() = β a 0 a = 0 a + 0 a 0 +0, 0

30 5.0 f() = f() = { + ( < 0) ( > 0) f() =, +0 f() = 0.0 f() f() a+0 a 0 f() a f() f() f() = f() = α f() = α a+0 a 0 a.3 () +0 () 0 (3) +0 (4) 0

31 .. 53 f() =, a+0 f() = a 0. f() = y y = +0 = 0 = O.4 () +0 () 0

32 54.. () A f() f() = y f() 0 y = f() 0 O f() α α f() f() = α = 0, = 0 f() a = ( ) = = ( ) = y O y = y =

33 () = 0 () 3 + = 0 (3) ( 3 ) = 3 ( ) =.5 () () (3) ( ) (4) ( + 3 ) () () () = = () =.6 () = () (3) (4) 4 +

34 56.6 () ( 4 + ) () ( + + ) () 8. () = t t () ( ( 4 + )( ) = + + ) = (4 + ) () = 4 + = + = = 4 () = t t ( + + ) = ( t t t) t ( t t t)( t = t + t) t t t + t (t t) t = t t t + t = t t t t + t = t t t t = t + t t + =

35 () ( + ) () ( )

36 58 B a > y = a y = log a a = a = 0 0 < a < a = 0 a = 0 < a < y O a > O y a > 0 < a < a > log a = log a = +0 0 < a < log a = log a = +0.8 ( ) () () (3) 3 log (4) log () + 3 () log () = 0 () + 3 = ( = log + 3 ) = log = log () 3 4 () (3) log +

37 A sin cos sin cos tan y y = sin O π π 3 π π 5 π y = cos tan y y = tan tan = π +0 tan = π 0 π tan π O π π π 3 π.3 sin = 0, 0 sin = 0, cos = 0 cos =.30 () sin () cos (3) π tan

38 60 () f() = α g() = β a a 5 = a f() g() α β 6 = a f() h() g() α = β h() = α a 6 f() h() g() f() < h() < g() 5 6 f() = f() g() g() =. sin 0 0 sin 0 sin = sin = sin = 0 0 sin = 0.3 () 0 cos () sin (3) cos

39 B sin.. 6 sin sin sin!0 = 0 < < π OAB tan A OB T O A OAB < OAB < OAT B T sin < < sin < < tan sin > 0 < sin < cos > sin tan > cos sin π < < 0 0 < < π > sin( ) > cos( ) > sin > cos cos = 6 0 sin 0 =

40 6.3 sin sin () () 0 0 sin 3 sin () = 0 0 sin () 0 sin 3 = 0 = 0 ( sin ) ( ) 3 3 sin 3 sin ( ) sin 3 sin 3 3 = 3 = 3 = = () 0 sin 3 () 0 tan (3) 0 sin 3 sin 5

41 cos 0 (cos + )(cos ) = cos = sin cos + cos 0 = 0 (cos )(cos + ) (cos + ) cos = 0 (cos + ) = sin 0 (cos + ) ( = sin ) 0 sin 0 = cos + + = 0.33 () 0 cos () 0 cos sin

42 64..4 A y y = f() f(a) f() a f() = f(a) a O a ( ).4 f() = y y = f() ( = ) y = f() = f() = f() = f() f() f() f() = +.34 f() f() = f(0) a f() = + 0 ( 0) a ( = 0) O

43 .. 65 [] [ ] f() = [] y = f() y f() = 4 3 f() =, f() = f() O f() = f() f() y a f() a f() = f(a) f(a) a f() = a y = f() = a O a y = f() a f() = f(a) a+0 f() = f(a) a 0 f() = a.5 f() = y = 0, f() = 0 +0 f() = f() +0 O y = f() = =

44 66 f() = a = a = a f() = [].35 f() = 0 () f() = [] () f() = ( + )[] (3) f() = 44 4 f() g() = a = a k f(), f() + g(), f() g(), f()g(), f() g() f() k g() g(a) 0 B { a < < b} { a b} { a } { < b} (a, b) [a, b] [a, ) (, b) (, ) (a, b) [a, b] f() f().6 () a sin cos (, ) () log a (0, ) (3) = (, ) (, )

45 .. 67 f() g() I I k k f(), f() + g(), f() g(), f()g() f() g() I g() = 0.36 () f() = + () f() = 3 + C.7 f() = sin [0, π] f() = π = 0, π 0 y = sin f() = cos () [0, π] () [ π, π] f() [a, b] y y = f() f(b) f(a) f(b) f(a) f(b) k y = k y = f() a < < b k f(a) O a b

46 68 f() [a, b] f(a) f(b) f(a) f(b) k f(c) = k a < c < b c y y = f() f(b) f() [a, b] f(a) f(b) f() = 0 a < < b a O f(a).4 cos = 0 0 < < π f() = cos f() [0, π] f(0) = 0 cos 0 = < 0 f(π) = π cos π = π + > 0 f(0) f(π) f() = 0 cos = 0 0 < < π.38 3 = 0 3 < < 4 b

47 () + () {log ( + 4) log } 5 () 0 cos () 0 tan sin (3) π + cos ( π)

48 70 6 f() = a f() = 3 ( ) a ( = ) = 0 4 () () 5 () 6 a = 3 [ () (3) 3 = ( )( + + ) ] = ( + + ) 7 f() = f( ) < 0 f(0) > 0

49 A () n ( n) () n n n n n(n ) + n + n n

50 7 3 P O y y y 3 P O 4 (4) a + 3 () () + (3) 0 (4) {log a (a + ) log a }

51 () 0 cos tan () 0 tan sin 3 6 log + = < <

52 74.3. B 7 {a n } a = 3, a n+ = 3 a n (n =,, 3,...) () a n = n + n + () {a n} 8

53 f() f() = f() = 0 r O θ AB AB C D () θ +0 AB AB CD () θ +0 AB O B C θ r D A

54 76 f() y = f() f() ( + ) + + ( + ) + n y O 7 () a 8 a r r = a + r a r 9 f() = a + b + c 0

55 () 0 () [ () = n(n + )(n + ) 6 = n (n + ) 4 () = (3n ) = (n ) 3 0 n = ( + ) n + n C + n C ( ) [ 4 3 3, (, y) 3 = y = ] () () (3) 0 (4) [ (3) = t 0 ] ] = t t 5 () () 6 [ f() = log + ] f() < 0 f() > 0 7 () [ () n = k a k = k + k + a k+ = ] ( k + ) = = 3 a k 3 k + 3( k + ) ( k + ) = k + k + k [ 3 a r 3 a r = a r = a ] + r a r = r <

56 78 [ 9 f() = f() = a + b + c f() = a + b + c f() = 0 a + b + c = 0 f() = [ = a a = ( )( + b + ) ( + )( ) c = b = b + 4 ] 0 () () 8r [ = 0 + AB = rθ AB = r sin θ CD = r ( cos θ = 0 f() = 0 0 f() = y + ) ] ] = + O

春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim n an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16,

春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim n an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16, 春期講座 ~ 極限 1 1, 1 2, 1 3, 1 4,, 1 n, n n {a n } n a n α {a n } α {a n } α lim an = α n a n α α {a n } {a n } {a n } 1. a n = 2 n {a n } 2, 4, 8, 16, 32, n a n {a n } {a n } 2. a n = 10n + 1 {a n } lim an

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