. 99 () 0 0 0 () 0 00 0 350 300 () 5 0 () 3 {a n } a + a 4 + a 6 + + a 40 30 53 47 77 95 30 83 4 n S n S n = n = S n 303 9 k d 9 45 k =, d =
99 a d n a n d n a n = a + (n )d a n a n S n S n = n(a + a n ) = n{a + (n )d} 300 (), 3, 8, 3, 8, 3, 8, (), 5, 8,, 4, 7, 0, 30 a n a n n 30 S n n S n n S n a n a n 0 S n 303 9 k +
99 () {a n} a n = + (n ) ( ) = n + 3 0 a 0 = 0 + 3 = 7 0 0 a 0 = 7 a 0 = 0+3 = 7 ( 7 7) = ( ) = 3 n S n a 0 + a + + a 0 = S 0 S 9 = 0 { + 9 ( )} 9 { + 8 ( )} = 3 () a d 0 S 0 = 00 0 (a + 9d) = 00 a + 9d = 0 S 0 = 350 0 (a + 9d) = 350 a + 9d = 35 a = 3 4, d = 3 300 () {a n } {a n } 0 a n = + 0(n ) = 0n a 0 = 0 0 = 88 5 {b n } b n = + (n ) 5 = 5n 7 {c n } c n = b n c 0 = b 0 = b 39 = 5 39 7 = 88 () {a n} 3 a n = + (n ) 3 = 3n a, a 4, a 6,, a 40 a a 40 0 a = 3 = 5, a 40 = 3 40 = 9 S S = 0 (5 + 9) = 40
30 {a n } a d a 53 = 47 a + 5d = 47 a 77 = 95 a + 76d = 95 d =, a = 57 a n = 57 + (n ) ( ) = n + 59 a n < 0 n + 59 < 0 n > 59 = 9.5 30 30 n{ ( 83) + (n ) 4} S n = = n(n 85) ( = n 85n = n 85 ) 85 4 8 85 4 = + 4 n = S n : S = ( 85) = 903 n{ ( 83) + (n ) 4} S n = = n 85n 83 4 a n a n = 83 + (n ) 4 = 4n 87 {a n } a n 0 a n S n a n 0 4n 87 0 n 87 4 =.75 n = S = 85 = 903 303 9 k + 45 (k + )( + 9) = 45 k = 47 49 9 d + (49 )d = 9 d = 6
. 304 a, a, a a = 305 3 a, b, c 6 44 a =, b =, c = a < b < c 304 305 a, b, c b a = c b = b = a + c
304, a, a a = + a a + a = 0 a > 0 a = + 3 305 a, b, c b = a + c { a + b + c = 6 a + b + c = 44 3 3b = 6 b = 3 { a + c = 4 a + c = 40 c a + (4 a) = 40 a = 6, a < b < c a =, c = 6
.3 306 a, b,, c, 8 a =, b =, c = 307 () 5 48 6 96 8 () 3 4 45( + ) 308 7, 77, 777, 7777, () n a n () n S n
306 a r n a n r n a n = a r n 307 () 5 6 a n a n S n a ( r n ) r \= S n = r na r = () r n 308 a = 7 a = 77 = 700 + 7 = 7 0 + 7 a 3 = 777 = 70000 + 700 + 7 = 7 0 4 + 7 0 + 7 a 4 = 7777 = 7 0 6 + 7 0 4 + 7 0 + 7 a n 0
306 {a n } (n =,, 3, 4, 5) a r a 3 = ar = a 5 = 8 ar 4 = 8 ar 4 ar = 8 r = 9 r = 3 a = 9 b = a = ar = 9 3 = 3, c = a 4 = ar 3 = 9 33 = 6, c, 8 c = 8 c = 36 c c = 6, 6, 8 3 b = 3, a = 9 307 () {a n } a r 5 a 5 6 a 6 a 5 = 48 ar 4 = 48 a 6 = 96 ar 5 = 96 ar 5 ar 4 = 96 48 r = r = a = 3 8 S 8 S 8 = 3{ ( )8 } ( ) = 56 = 55
() r n 3 4 4 = 3r n r n = 8 45( + ) 45( + ) = 3( rn ) r r = 5( + )( r) 8 r = ( ) n = ( ) 7 n = 8 308 () a n n r n = 5( + )( r) r = 5 + 4 7 + 5 = a n = 777 7 = 7 00 n + 7 00 n + + 7 00 + 7 7 00 n a n = 7(00n ) = 3 00 (00n ) () n S n S n = n a k = n 3 k= k= (00k ) = 3 n k=00 k 3 = 3 00(00n ) 3 00 n = 00n+ 00 363 3 n = 00n+ 99n 00 363 n k=
.4 309 < a < b, a, b a, b, ab a, b 30 3 A, B, C 3 α, β, αβ (α < 0 < β) α + β = β α = αβ = 309 a, b, c b a = c = b b = ac 30 A, B, C A < B < C A > B > C A, B, C B = AC 3 α < 0, β > 0 αβ < 0 3 α, β, αβ +
309, a, b a = + b a, b, ab (b) = a ab b = a ( b \= 0) a 4a + = 0 a = + ( a > ) b = ( + ) = 3 + 30 A, B, C B = AC A C X 3 A, B, C X, B, π B = π X 0 < X < B < π X + B = π B = π ( ) π B 4B + πb π = 0 B = π + 5π 4 A < B < C ( A < B < π ) = π 5 + B = A > B > C ( ) π > B > C = B π 5 = 3 α < 0 < β αβ < 0 α, β, αβ 3 β β = α αβ β = α ( β \= 0) α + β = α < 0 α + α = 0 α = 5 β α = + 5, αβ = + 5, β = α = 3 + 5
.5 3 5% 000 000 log 0 = 0.30 log 0.05 = 0.0 3 + = + 0.05 =.05
3 + 5 =.05 000 00 n 000 000 0 4.05 n > 000 0 4.05 n > n log 0.05 > log 0 n > log 0 log 0.05 = 0.30 0.0 = 4.3 000 5
.6 33, 4, 8, 3, 9, 6, 34, 30 34,, 0, 4, 0, 8,, 40, 08 0 33 {a n} {b n} b k = a k+ a k n b k = n (a k+ a k ) k= k= n a n = a + n = (a a ) + (a 3 a ) + (a 4 a 3) + + (a n a n ) + (a n a n ) = a + a n b k k= 34 0, 4, 0, 8
33 {a n } {b n } a n : 4 8 3 9 6 34 b n : 3 4 5 6 7 8 {b n} 3 b n = n + n a n = + n (k + ) = n + 3 n n = k= a 30 = 30 + 3 30 = 494 34 {a n } {b n } a n : a 0 4 0 8 a 7 40 b n : b b 4 6 8 b 6 b 7 {b n} b =, b = 0, b 6 = 0, b 7 = b = 0 a, b = a ( ), b 6 = a 7 8, b 7 = 40 a 7 a =, a + = 0, a 7 8 = 0, 40 a 7 = a = a 7 = 8 b n = 0 + (n ) = n n a n = + n (k ) = n 3n n = k= a n = 08 n 3n 08 = 0 (n + 9)(n ) = 0 n n = 08 0 0 n= (n 3n) = 0 4 6 3 0 = 40
.7 35 () () n k=n+ k n (k 4) 3 k=3 36 (), +, + + 3, + + 3 + 4, 5 3 4 () n (n ) + (n ) + + (n ) + (n ) (3) + 3 + 3 4 + 4 5 + + n(n + ) = n 37 (k + ) 5 k 5 = 5k 4 + 0k 3 + 0k + 5k + n k= k 4 38 S n = n kx k k=
n = n, k= n k = k= n k = k= n(n + ) n(n + )(n + ) 6 35 () S n = n k k=,, n k 3 = n (n + ) 4 k= S n = + + + n + (n + ) + (n + ) + + n S n = + + + n S n S n = (n + ) + (n + ) + + n = n k=n+ () k = 3, 4,, n n (k 4) 3 k=3 36 () n + + 3 + + n n () (3) 37 n k, k= k = ( + + 3 + + n) n n k 3 k= 38 {( ) ( )} S n r S n rs n a k = k b k = x k {a n } {b n } x S n = n k= a k b k
35 () n k=n+ k = n k= k n k= k = n (n + ) n(n + ) = 4n4 + n n () n ( + ) k = k=n+ = n(n )(n + n + ) = (n n){(n + ) + n } n (k 4) 3 = (3 4) 3 + (4 4) 3 + (5 4) 3 + + (n 4) 3 k=3 = ( ) 3 + 0 3 + 3 + 3 + + (n 4) 3 = + n 4 = + (n 4) (n 3) 4 = (n )(n 5)(n 7n + 4) 4 = (n 7n + ) 4 k 3 k= 36 () {a n } a n n n(n + ) a n = + + 3 + + n = n(n + ) a n = = (n + ) n 5 5 k = k=a + ) = k=(k ( 5 6 ) + 5 = 75 () (n ) + (n ) + + k(n k) + + (n ) k(n k) = n n k n k = n = n k= k= k= (n )n (n )n(n ) 6 = (n )n(n + ) 6
(3) + 3 + 3 4 + 4 5 + + n(n + ) = n k(k + ) = k= = 6 n k= (k + k) = 6 n(n + )(n + ) + n(n + ) n(n + )(n + + 3) = n(n + )(n + ) 3 37 k (k + ) 5 k 5 = 5k 4 + 0k 3 + 0k + 5k + k =,, 3,, n n {(k + ) 5 k 5 } = n (5k 4 + 0k 3 + 0k + 5k + ) k= k= (n + ) 5 5 = 5 n k 4 + 0 n k 3 + 0 n k + 5 n k + n k= k= 5 n k 4 = (n + ) 5 0 n k 3 0 n k 5 n k n k= k= k= k= = (n + ) 5 0 n (n + ) n(n + )(n + ) 0 4 6 n(n + ) 5 = n 5 + 5 n4 + 5 3 n3 6 n n k 4 = 5 n5 + n4 + 3 n3 k= k= 30 n k= n 38 x = S n = n k= k = n(n + ) x \= S n = + x + 3x + + (n )x n + nx n xs n = x + x + + (n )x n + (n )x n + nx n ( x)s n = + x + x + + x n + x n nx n = xn x nxn = xn nx n ( x) x S n = (n + )xn + nx n+ ( x)
.8 39 3 + 3 5 + 5 7 + + 7 9 + 9 = 30 4 7 + 4 7 0 + + (3n )(3n + )(3n + 4) () 3 50 k= k + + k 39 n k= (k )(k + ) = n k= A k + B k + ( A k + = (A + B)k + (A B) (k )(k + ) B ) k + k =,,, n k ( AB = B A A ) B 30 (3k )(3k + )(3k + 4) = A (3k )(3k + ) + B (3k + )(3k + 4) ( ABC = C A AB ) BC 3
39 3 + 3 5 + 5 7 + + 7 9 + 9 = 0 (k )(k + ) = A k + A k + B k + B k + = (k + )A + (k )B (k )(k + ) k = k= (k )(k + ) (A + B)k + (A B) (k )(k + ) A + B = 0 A B = A =, B = 0 k= (k )(k + ) = 0 ( ) k= k k + = { ( ) ( + 3 3 ) ( + + 5 7 ) ( + 9 9 ) } = ( ) = 0 = 0 ( AB = B A A ) B ( ) (k )(k + ) = (k + ) (k ) k k + = ( ) k k +
30 (3n )(3n + )(3n + 4) = A (3n )(3n + ) + B (3n + )(3n + 4) (3n + 4)A + (3n )B 3(A + B)n + (4A B) = = (3n )(3n + )(3n + 4) (3n )(3n + )(3n + 4) 3(A + B) = 0 4A B = A = 6, B = 6 = n k= (3k )(3k + )(3k + 4) = { } n 6 k= (3k )(3k + ) (3k + )(3k + 4) = [ ( 6 4 ) ( + 4 7 4 7 ) + 7 0 { }] + (3n )(3n + ) (3n + )(3n + 4) = { } 6 4 n(3n + 5) = (3n + )(3n + 4) 8(3n + )(3n + 4) ABC = ( C A AB ) BC (3n )(3n + )(3n + 4) { } = (3n + 4) (3n ) (3n )(3n + ) (3n + )(3n + 4) = { } 6 (3n )(3n + ) (3n + )(3n + 4) 3 k + + k = k + + k k + k k + k = k + k 50 = 50 ( k + k) k + + k k= k= = ( ) + ( 3 ) + + ( 50 49) + ( 5 50) = 5
.9 3 {a n } n S n S n = n n a n = 33 n 4 n 3 5 n 3 S n a n { S n = a n = S n S n n n S n = a + a + + a n + a n ) S n = a + a + + a n S n S n = a n 33 S n = 4 n a n n a n
3 a = S = = 0 n a n = S n S n = (n n) {(n ) (n )} = n a = 0 a n = n (n ) 33 {a n } n S n = 4 n n = a = S = 4 n a n = S n S n = 4 n 4 n = 3 4 n {a n } 3 5 {b n } { n = b = a = 4 n b n = a n = 3 4 n = 3 6 n
.0 34 () 4 5 () 43 35 00 3 34 () 4 5 = 0 0,,,, 0 () 43 = 4 3 3 0 3 0, 0 3, 0 3, 0 3 3, 3 0, 3, 3, 3 3,, 4 3 0, 4 3, 4 3, 4 3 3 35 A : B : 3 00 A B : 3 A B : 6 A B n(a B) = n(a) + n(b) n(a B) A B
34 () 4 5 = 0,,, 3,, 0 + + + 3 + + 0 = = 047 () 43 = 4 3 3 i 3 j (0 i 4, 0 j 3) (i, j) (4 + ) (3 + ) = 0 ( + 3 + 3 + 3 3 ) + ( + 3 + 3 + 3 3 ) + 3 ( + 3 + 3 + 3 3 ) + 4 ( + 3 + 3 + 3 3 ) = ( + + 3 + 4 )( + 3 + 3 + 3 3 ) = (4 ) 34 3 = 00 i 3 j ( i 4, 0 j 3) 4 3 i 3 j = 4 ( i 3 3 j) = 4 ( ) i 34 = 40 4 i i=j=0 i= j=0 i= 3 i= = 40 (4 ) = 40 30 = 00 35 00 = 50 50 00 3 = 33 3 33 00 6 = 6 4 6 6 3 50 + 33 6 = 67 3 00 67 = 33 50( + 00) = + 4 + 6 + + 00 = = 550 33(3 + 99) 3 = 3 + 6 + 9 + + 99 = = 683 6(6 + 96) 6 = 6 + + 8 + + 96 = = 86 00( + 00) 00 = + + 3 + + 00 = = 5050 5050 (550 + 683 86) = 633
. 36 n 3 n 3 n 3 3 3 3 3 n........ n n n 3 n 37 n,, 3,, n S 36 ( + + + n) ( + + + n ) + { + 3 + + (n ) n} (a + b + c + ) a + b + c + 37 3 4 n 3 4 n 3 4 3 n...... (n )n
36 n ( + + + n) { } n(n + ) = ( + + + n) = = n (n + ) 4 k k, k,, k n = n (k + k + + k n) = ( + + + n) n k = k= n(n + ) 37 S n(n + ) = n (n + ) 4 k= ( + + 3 + + n) = S + ( + + 3 + + n ) S = ( + + 3 + + n) ( + + 3 + + n ) ( n ) = n = k k= k k= { } n(n + ) n(n + )(n + ) 6 = n(n + ){3n(n + ) (n + )} = n(n + )(n )(3n + ) S = n(n + )(n )(3n + ) 4 ij i < j n ( ) { S = n j ij = n j j i = n j } j(j ) j= i= j= i= j= = n j=(j 3 j ) = { n } (j 3 j ) ( 3 ) j= = { 4 n (n + ) } n(n + )(n + ) 6 = n(n + )(n )(3n + ) 4
. 38 x y (x, y) () x 0, y 0, x + y 0 () y 0, y x, x + y 0 39 n () x + y n, x 0, y 0 (x, y) ( ) n + n + () x + y + z n, x 0, y 0, z 0 (x, y, z) ( ) n 3 + n + n + 6 38 () x = y = () x + y = 0 x = k y = 0 k k =, y = k x = 0 k (x, k) 39 () y x = k (k :, 0 k n) x y = l (l :, 0 l n) () () xy z = l (l :, 0 l n)
38 () k = 0,,, 0 x = k, 0 y 0 k 0 k + = k 0 ( k) = { + ( 0)} k=0 = 3 x 0, y 0, 0 (x + y) 0 z = 0 (x + y) x + y + z = 0, x 0, y 0, z 0 y 0 0 k (x, y, z) 0 ( : (x, y, z) = (,, 7) ) O k 0 x 3H 0 = C 0 = C = 3 n r nh r = n+r C r () A(4, 8) B(0, 0) C(4, 0) OAC AC y k = 0,,, 3 x = k, 0 y k k + + 3 + 5 + 7 = 6 8 ABC l = 0,,,, 8 y = l, 4 x 0 l (0 l) 3 = 7 l 8 (7 l) = 9 {7 + (7 6)} y l=0 6 + 8 = 97 = 8 8 l 0 A C O 4 y = x x + y = 0 B 0 y = x 0 A x + y = 0 C B O 4 5 6 7 0 l 0 x x
39 () x = k, 0 y n k (0 k n) n k + (x, y) n (n k + ) k=0 = (n + ) + n + (n ) + + 3 + + = (n + )(n + ) = (n + 3n + ) () z = l (0 l n) x + y n l, x 0, y 0 z = l () (n l + )(n l + ) (x, y, z) n (n l + )(n l + ) l=0 y n n k O y n l k n z = l = {(n + )(n + ) + n(n + ) + O n l + 3 4 + 3 + } = n+ + ) = j=j(j { } (n + )(n + )(n + 3) (n + )(n + ) + 6 = (n + )(n + ) {(n + 3) + 3} = (n + )(n + )(n + 3) 6 6 = 6 (n3 + 6n + n + 6) x x () n x y 0, x 0, y 0 z = n x y x + y + z = n, x 0, y 0, z 0 (x, y, z) (n + )(n + ) 3H n = n+ C n = n+ C = = (n + 3n + ) () w = n x y z x + y + z + w = n, x 0, y 0, z 0, w 0 (x, y, z, w) (n + 3)(n + )(n + ) 4H n = n+3c n = n+3c 3 = 3 = 6 (n3 + 6n + n + 6)
.3 330,, 3, (), (, 3, 4), (5, 6, 7, 8, 9), (0,,, 3, 4, 5, 6), n n n n 6 3 () n n () 365 (3) n n c n c + c + + c M M 33 3 5 4 5 7 4 7 6 7 9 4 9 6 9 8 9 () 4 5 () 00 ( i ) n (ii) n (iii) n (iv) n 330 n m a n,m () n a n, n + () a n,m = 365 a n, 365 < a n+, (3) c n = a n,n 33
330 (), (, 3, 4), (5, 6, 7, 8, 9), 3 (0,,, 3, 4, 5, 6), 4 () n m a n,m n n n (n ) n (k ) = (n ) n (n ) = (n ) k= n a n, = (n ) + = n n + n = () 365 n n 365 < n + () (n ) + 365 < n + 9 = 36 0 = 400 (0 ) + (= 36) 365 < 0 + (= 40) n = 0 0 36 365 365 36 + = 4 365 0 4 (3) n n c n = a n,n = (n ) + n = n n + c + c + + c M = M c k k= = M (k k + ) k= = 6 M(M + )(M + ) M(M + ) + M = M{(M + )(M + ) 3(M + ) + 6} 6 = 3 M(M + )
33 3 5 4 5 7 4 7 6 7 9 4 9 6 9 8 9 3 5, 4 5 7, 4 7, 6 7 9, 4 9, 6 9, 8 9 3 4 n n n m a n,m () a n,m = m n + 4 5 = 7 7 + ( + + 3 + 4 + 5 + 6) + = 6 7 + = 3 () 00 n n + + + (n ) < 00 + + + (n ) + n (n )n < 00 n(n + ) (n )n < 00 n(n + ) 3 4 = 8, 4 5 = 0 n = 4 3 + + + 3 = 3 4 = 9 00 4 00 9 = 9 a 4,9 = 9 4 + = 8 9
7. 33 n 3 + 3 + 3 3 + + n 3 = n (n + ) 4 333 5 n n > n 334 a = 3, a n+ = a n (n =,, 3, ) {a n } a n n 33 n P (n) (I) P () (II) P (k) P (k + ) n P (n) 333 (I) n = 5 (II) 5 k 334 n n n
7 33 3 + 3 + 3 3 + + n 3 = n (n + ) 4 (I) n = = 3 =, = ( + ) = 4 (II) n = k k 3 + 3 + + k 3 + (k + ) 3 = k (k + ) 4 + (k + ) 3 = (k + ) {k + 4(k + )} 4 = (k + ) (k + ) 4 = (k + ) {(k + ) + } 4 n = k + (I) (II) n 333 (I) n = 5 5 = 3 > 5 = 5 (II) n = k ( 5) k > k k+ (k + ) = k k k > k k k ( ) = k k = (k ) k 5 (k ) (5 ) = 4 > 0 k+ > (k + ) n = k + (I) (II) 5 n n > n
7 334 a = 3 ( a = = ) = 3 a 3 5 ( a 3 = = 3 ) = 5 a 5 7 ( a 4 = = 5 ) = 7 a 3 7 9 a n = n n + (I) n = + = 3 (II) n = k a k = k k + a k+ = = a k k = k + (k + ) = k + 3 (k + ) + k + n = k + (I) (II) n a n = n n +
7. a n+ = a n + q(n) 335 {a n } a =, a n+ = a n 6n + 3 (n =,, 3, ) a n n a n n = 336 a = 0 n(n ) a n+ a n = + (n =,, 3, ) {a n } 335 336 a n+ = a n + f(n) a n+ a n = f(n) {a n} f(n) n =,, 3,, n (a a ) + (a 3 a ) + (a 4 a 3 ) + + (a n a n ) = f() + f() + f(3) + + f(n ) a n a = n f(k) (n ) k= a n = a + n f(k) (n ) k= n =
7 335 a n+ = a n 6n + 3 a n+ a n = 6n + 3 n a n = a + n ( 6k + 3) = 6 k= (n )n + 3(n ) = 3n + 6n n = ( = 3 n 8 ) + 8 3 3 n = 3 a 3 = 9 336 n { a n = a + n k(k ) k= = { (n )n(n ) 6 = n = n } + = n k= (n )n {n(n ) 3n} + n {n(n ) 3n + } = 6 (n )(n n + 6) a = 0 n = a n = 6 (n )(n n + 6) (n ) (k k) + (n ) } + n
7.3 a n+ = pa n + q 337 a = 3, a n+ = 3a n + (n =,, 3, ) a n 338 a 0 =, a n = 3 a n + 3 (n =,, 3, ) {a n } 339 {a n } n S n S n = a n + 3n + (n =,, 3, ) a n 337 a n+ = pa n + q {a n } p = a n+ = a n + q q q = 0 a n+ = pa n p a n+ = pa n + q { an+ = pa n + q α = pα + q α a n+ α = p(a n α) {a n α} a α p 338 a 0 n = 0 339 { S n = a n = S n S n n
7 337 a n+ = 3a n + a n+ + ( = 3 a n + { a n + a + = 3 + = 7 ( α = 3α + α α = ) ) } 3 a n + = 7 3n a n = 7 3n 338 a n = ( 3 a n + 3 α = 3 α + 3 α α = 9 ) a n 9 = ( a n 9 ) 3 { a n 9 } 3 a 0 9 = 9 = 5 a n 9 = 5 ( ) n 3 a n = 5 3 n + 9 339 a = S = a + 5 a = 5 a n+ = S n+ S n = { a n+ + 3(n + ) + } ( a n + 3n + ) = a n+ + a n + 3 a n+ = a n + 3 a n+ = a n + 3 (α = α + 3 ) α α = 3 a n+ 3 = (an 3) {a n 3} a 3 = 5 3 = a n 3 = ( ) n = n a n = 3 n
7.4 a n+ = pa n + q 340 {a n }, {b n } a = 6 a n+ = () b n+ b n a n 6a n + 7, b n = a n (n =,, ) () {a n }, {b n } a n, b n 34 {b n } b = b n+ = 5 b n (n =,, 3 ) b n n 340 a n+ = ran p q r pa n + q 30 34 b n+b n bn a n b n = b n log a a
7 340 () a = 6 > 0 a n > 0 b n = = 6a n + 7 = 7 a n+ a n a n b n+ = 7b n + 6 a n + 6 () () β = 7β + 6 β β = b n+ + = 7(b n + ) {b n + } 7 b + = a + = 6 + = 7 b n + = 7 7 n b n = 7 n a n = b n = 7 n 34 b n+ = 5 b n (> 0) 5 log 5 b n+ = log 5 5 b n = log 5 5 + log 5 b n = + log 5 b n a n = log 5 b n a = log 5 b = log 5 = 0 a n+ = a n + (α = ) α + α α = a n+ = (a n ) {a n } a = ( ) a n = n = n a n = n = n log 5 b n = n b n = 5 n
7.5 a n+ = pa n + q(n) 34 a = 3, a n+ = a n + n (n =,, 3, ) b n = a n+ a n {b n } b n = {a n } a n = 343 a =, a n a n = n (n =, 3, ) {a n } n f(n) = pn + q a n + f(n) = {a n + f(n )} (n =, 3, ) p =, q = a n = (n =,, ) 344 a n a =, a n+ = 3a n + n (n =,, 3, ) ( ) 345 a = a n+ = 4a n + n (n =,, 3, ) 3 ( ) {a n } α b n = a n + α n (n =,, 3, ) 3 α = {b n } { ( ) n } a n = n (n =,, 3, )
7 34 p q n p = a n+ a n = q(n) {a n} {q(n)} p \= 0, a n+ a n = b n {a n } 343 a n+ = pa n + q(n) f(n + ) = pf(n) + q(n) n f(n) a n+ f(n + ) = p{a n f(n)} {a n f(n)} p 344 q(n) = r n n r n+ a n+ = pa n + q(n) a n+ r n+ = p r an r n + r b n+ = Ab n + B p n+ a n+ = pa n + q(n) a n+ p n+ = a n p n + q(n) p n+ { an p n } { } q(n) p n+ ( ) 345 343 f(n + ) = 4f(n) + n f(n) 3 a n+ f(n + ) = 4{a n f(n)} 4 {a n f(n)} ( ) f(n) = α n 3
7 34 b n = a n+ a n = (a n + n ) {a n + (n ) } = a n a n + = (a n a n ) + = b n + α = α + α α = b n + = (b n + ) {b n + } b + = a a + = (a + ) a + = a + = 4 b n + = 4 n = n+ b n = n+ {a n } {b n } n a n = a + n b k = 3 + n ( k+ ) = 3 + 4(n ) (n ) k= = n+ n k= n = a n = n+ n
7 343 a n + f(n) = {a n + f(n )} f(n) = pn + q a n + pn + q = {a n + p(n ) + q} = a n + pn p + q a n a n = pn + ( p + q) a n a n = n n =, 3, p = p + q = 0 p =, q = f(n) = n + {a n + f(n)} a + f() = + + = 4 a n + f(n) = 4 n = n+ a n = n+ f(n) = n+ n a n+ a n = n +, a n a n = n (a n+ a n ) (a n a n ) = b n = a n+ a n b n = b n +, b = a a = (a + ) a = 4 = 3 β = β + β β = b n + = (b n + ) {b n + } b + = 3 + = 4 b n + = 4 n = n+ b n = a n+ a n = n+ n a n = a + n ( k+ ) = + 4 n k= (n ) = n+ n n =
7 344 a n+ = 3a n + n n+ b n = a n n a n+ n+ = 3 an n + b n+ = 3 bn + (α = 3 α + ) α α = b n+ + = 3 (b n + ) {b n + } 3 b + = a + = + = 3 b n + = 3 ( ) 3 n ( ) b n = 3 n = 3 n n n a n = n b n = 3 n n 3 n+ c n = an 3 n a n+ 3 n+ = a n 3 n + n 3 n+ c n+ = c n + n 3 n+ { {c n} n c n = c + n = 3 k= k 3 k+ 3 n+ } n = a 3 + n ( ) { k = + n ( ) } k 3 k= 3 3 k= 3 = { + ( ) 3 3 + ( ) } + + n 3 3 ( ) n 3 ( ) = n = 3 n n 3 3 n 3 a n 3 n = 3n n 3 n a n = 3 n n n = a n+ + n+ = 3(a n + n ) a n + n = (a + ) 3 n = ( + ) 3 n = 3 n a n = 3 n n
7 345 {b n } ( ) b n = a n + α n ( ) a n = b n α n 3 3 ( ) b n+ = a n+ + α n+ ( ) a n+ = b n+ α n+ 3 3 ( ) b n+ α n+ { ( ) = 4 b n α n } ( ) + n 3 3 3 ( b n+ = 4b n + 4α + α ) ( ) n 3 3 ( b n+ = 4b n + ) ( ) 3 α n 3 α = 3 3 α = 0 {b n} b = a + 3 3 = 4 b n = 4n = 3 4n a n = 3 4n 3 ( ) n 3 a n = 3 { 4 n ( ) n } 3
7.6 3 346 {a n } a =, a =, a n+ a n+ 3a n = 0 () b n = a n+ + a n c n = a n+ 3a n b n, c n n () a n n 347 {a n } a = a = 6 a n+ = 6a n+ 9a n (n =,, 3, ) () b n = a n+ 3a n {b n } () {a n } 346 a n+ + pa n+ + qa n = 0 a n+ αa n+ = β(a n+ αa n) {a n+ αa n} β a n+ (α+β)a n+ +αβa n = 0 α+β = p αβ = q α, β α, β x + px + q = 0 347 a n+ + pa n+ + qa n = 0 x + px + q = 0
7 346 () a n+ a n+ 3a n = 0 x x 3 = 0 x (x 3)(x + ) = 0 x =, 3 a n+ + a n+ = 3(a n+ + a n ) b n+ = 3b n a n+ 3a n+ = (a n+ 3a n ) c n+ = c n {b n } 3 b = a + a = + = 3 b n = 3 3 n = 3 n a n+ + a n = 3 n 3 {c n } c = a 3a = 3 = c n = ( ) ( ) n = ( ) n a n+ 3a n = ( ) n 4 () 3 4 4a n = 3 n ( ) n a n = 3n ( ) n 347 () x = 6x 9 x (x 3) = 0 x = 3 a n+ 3a n+ = 3(a n+ 3a n) b n = a n+ 3a n b n+ = 3b n {b n} 3 b = a 3a = 3 b n = 3 n () () a n+ 3a n = 3 n 3 n+ a n+ 3 n+ a n 3 n = 3 an+ an n+ = 3 3 n + { } 3 an 3 n 3 a 3 = 3 a n 3 n = 3 + (n ) 3 = n 3 4 a n = n 3 n
7.7 348 {a n }, {b n } { an+ = a n + b n a =, b =,, n =,, 3, b n+ = a n + b n () a n + b n n () a n n 349 {a n }, {b n } a =, b =, a n+ = a n + 3b n, b n+ = a n + b n () c n = a n + kb n {c n } k () () k d n = a n kb n {c n }, {d n } (3) a n, b n { an+ = pa n + qb n b n+ = ra n + sb n α a n+ αb n+ = β(a n αb n ) α, β {a n αb n } β α x x = px + q rx + s 348 { an+ = pa n + qb n b n+ = qa n + pb n r = q s = p α = ± 349 k = k + 3 k + k = ± 3
7 348 () a n+ + b n+ = (a n + b n ) + (a n + b n ) = 3(a n + b n ) {a n + b n } 3 a + b = + = 3 a n + b n = 3 3 n = 3 n () a n+ b n+ = (a n + b n) (a n + b n) = (a n b n) {a n b n} a b = = a n b n = ( ) ( ) n = ( ) n + a n = 3 n + ( ) n a n = 3n + ( ) n 349 () c n+ = a n+ + kb n+ = (a n + 3b n) + k(a n + b n) c n+ = ( + k)a n + (3 + k)b n {c n} r c n+ = rc n c n+ = ra n + rkb n + k = r 3 3 + k = rk 4 3 4 3 + k = ( + k)k k = 3 k = 3 ( k > 0) () k = 3 3 r = + 3 {c n} {a n + 3b n} + 3 c = a + 3b = + 3 c n = ( + 3) ( + 3) n = ( + 3) n 5 d n = a n 3b n () k = 3 {d n } {a n 3b n } 3 d = a 3b = 3 d n = ( 3) ( 3) n = ( 3) n 6 (3) 5 a n + 3b n = ( + 3) n 7 6 a n 3b n = ( 3) n 8 7 + 8 7 8 3 a n = ( + 3) n + ( 3) n, b n = ( + 3) n ( 3) n 3
7.8 350 a = 0, a n+ = a n + 3 (n =,, 3, ) {a n } () b n = b + a n+ b n n () a n n 35 a = 4, a n+ = 3a n 4 a n () n a n \= () b n = (n =,, 3, ) {a n } a n {b n} (3) {a n } a n+ = pan + q ra n + s x x = px + q { } rx + s α a n α 350 x = x + 3 35 x = 3x 4 x x =, x =
7 350 () b n+ = = + a n+ = b n+ = b n + a n + 3 a n + + = b n + = a n + 3 a n + = + (a n + ) a n + () b n+ = b n + α = α + α α = b n+ + = (b n + ) {b n + } b + = + = b n + = n = n + a b n = n + a n = = b n n a n = n = n n 35 () (I) n = a = 4 \= (II) a n \= a n+ = 3an 4 a n = an a n \= 0 a n+ \= (I) (II) n a n \= () a n+ = a n a n a n+ = a n a n b n = a n b n+ = b n + = (an ) + a n = + {b n } b = b n = + (n ) = n (3) b n = a n a n = = b n n a n = n + = = n 4n n a n a = 4 =
7.9 35 n p n () p () p n+, p n p n+ = 5 6 p n + 6 ( p n) (3) p n (n =,, 3, ) 353 p p p 0 < p < n Q n n () Q Q 3 Q 4 p () Q n+ Q n Q n+ p (3) p = 3 7 Q n n 354 A B C 3 B A B B C n A B C p n, q n, r n () n =, p n, q n, r n () p n, q n, r n n
7 4 n n + 35 n n + 353 n + Q n Q n+ 354 3 = q n+ = p n + qn n p n q n n + p n+ q n+ r n r n+
7 35 () p = 5 6 () n + n n + n n + p n+ = p n 5 6 + ( p n) 6 = 5 6 p n + 6 ( p n) (3) () () p = 5 6, p n+ = 3 p n + ( α = 6 3 α + 6 α α = ) p n+ = ( p n ) 3 { p n } 3 p = 5 6 = 3 p n = ( ) 3 n p n = ( ) 3 3 n + 3 353 () Q Q = p Q 3 3 Q 3 = ( p)p Q 4 3 4 Q 4 = ( p)p () ( i ) 3 n p( p)q n (ii) n + ( p)q n+ Q n+ = ( p)q n+ + p( p)q n (3) p = 3 7 Qn+ = 4 Qn+ + 7 49 Qn ( t = 4 7 t + 49 t t = 7, 6 7 Q n+ + 7 Qn+ = 6 (Q n+ + ) 7 7 Qn Q n+ 6 7 Qn+ = (Q n+ 6 ) 7 7 Qn )
7 {Q n+ + } 7 Qn 6 7 Q n+ + ( 7 Q n = Q 3 + ) ( ) 7 Q 6 n ( ) = 9 7 49 6 n 7 { Q n+ 6 } 7 Q n 7 Q n+ 6 ( 7 Q n = Q 3 6 ) ( 7 Q ) n ( = 9 7 49 ) n 7 Q n = 9 { ( ) 6 n ( ) } n 56 7 7 354 () A C p =, q = 0, r = A ( ) p = = 4 B ( ) ( ) q = + = C ( ) r = = 4 () n + B n A n + n C n + q n+ = p n + r n = (p n + r n ) () = p n + q n + r n = { q n 3 B q n+ = ( q n) = q n + q n+ 3 = } q 3 = 3 q n 3 = ( 3 ) n q n = { ( ) } n 3 A C ( q n ) 3 A C B p n = r n p n = r n = q n = { ( ) } n 3 B A C B