MÉTHODES ET EXERCICES

J.-M. MONIER I G. HABERER I C. LARDON MATHS PCSI PTSI MÉTHODES ET EXERCICES 4 e édition

Création graphique de la couverture : Hokus Pokus Créations Dunod, 2018 11 rue Paul Bert, 92240 Malakoff www.dunod.com ISBN 978-2-10-077661-0

K n,p(k) n(k) n(k) n,(k) n,(k) n(k) n(k)n(k) F F F F F (un)n N n N, un+2 = un R F R 3 u =(1, 1, 0) v =(1, 0, 1) w =(1, 2, 1) F = (u, v, w). F a =(an)n N, b =(bn)n N F a0 =1, a1 =0, b0 =0, b1 =1, F = (a, b) F R N F F F (u, v, w) F (u, v) (u, v, w) 2u v =2(1, 1, 0) (1, 0, 1) = (1, 2, 1) = w. (u, v) F B F (F )= (B) (F + G)+ (F G) = (F )+ (G).

P, Q K[] (P + Q) ( (P ), (Q) ) (P ) (Q) P K[] (P )= (P ) 1 A K[] ( 3 +1)A =0 A =0 a, b K, P K[] P (a) =P (b) =0 ( a)( b) P n n N P (k) (a),a K, P Kn[] P () = ( a) k k! C[] C k=0 P R[] R[] (S, P ) C 2 S P 2 SX + P P R[] 0 un ( n 1 n 2 3 + 1 n n n) n n2 +3n +2 n! n + 1 2 n 2 +3n +1 n n 1 ( n n 2 1+ 2 ) 1 n n n. an R+, n0 un = an 1, vn = an an 1, wn =, xn = an. 2 1+an n N 1 un = n n +1+(n +1) n. n N, un = 1 1. n n +1 n1 un + an n=1 un. 1 a ]1 ; + [, a +1 = 1 a 1 2 a 2 1. + 2 n x ]1 ; + [ x 2n +1. (φn)n0 φ0 =0,φ1 =1 n N, φn+2 = φn+1 + φn. n N φn n + 2 n. n=0 φn n=0 1 4 (g u) (f u) y (g) (u) y (f) (g) (u) (f) x (g u) x (f u) (g v) (g u) y (v) g(y) (g u) y (u)+ (g) (v) (u)+ (g) z (g v) x E z =(g v)(x) (t, w) E 2 v(x) =u(t)+w z (g u) p f a 0 p 2 = p. f e = f 2 =0 = (f) (f), (f)+ (g) =E, (f) (g) ={0}. x E f ( x f(x) ) (f) (g). f g (f +f ) (f)+ (f ) (f + f, f ) (f, f ). (f 2 ) (f) (f 2 ) (f) {0} (f 2 ) (f) R 3, (f 2 ) (f) y (g f) (g f) f(y) =0 y =0 x E x (g)+ (f) (f)+ (g) x (g f)+ (g f) (p + q) 2 =(p + q) (p + q) =p 2 + p q + q p + q 2. p q p q q p. p q+q p =0, p q (f g +2e) f = e. E u, v L(E) u v = e, v u = e. g (f) g (f) g (f) : (f) G, y g(y). g (f). x E {0}, λx K f(x) =λxx, λx x λx x (x, y) ( E {0} ) 2 f(x), f(y), f(x + y) (x, y)

F f f(a) a = 1, b=1,f: t t 2 b 1 1 [ t f 2 ] 1 (t) t = 2t t =2 t t =2 2 a u(x) 1 f(b) f(a) =1 2 ( 1) 2 =0 1. x 2 f : x (x2 ) 2 2x ( x2 )1 = 2x x4 x2. 0 0 =1 u, v : I R C 1 I f : J R J u(i) J v(i) J v(x) G : I R, x f(t) t C 1 I x I, G (x) =f ( v(x) ) v (x) f ( u(x) ) u (x). π/2 x 2+ 3 x x = 0 0 1 1 2+t 3 t. x t t = x x t = x B y x y + 1 3 =0, y = x + 1 3. 1 y = x + 1 x = 1 E(x, y) Δ x x 3 3 y =0 y =0. x 2 1 1 2 F (x, y) E D2, F D2 F D2 y =2x +2 (EF) D2 EF D2 (x + 1 ) 1 3,y (1, 2) y =2x +2 x = 13 15 x + 1 3 +2y =0 y = 4 y =1 x 15. x =1 A(x, y) D1 D3 y = 1 2 x 1 y =0 ( 2 y =1 x x = 1 F 13 15, 4 ) D1 x + y 1=0 15 3 B(x, y) D1 D2 y =2x +2 y = 4 13 d(f, D1) = 15 + 4 15 1 = 8 3 1 2 +1 2 5 2 = 4 2 1, 13... 5 y =2x +2 x = 5 3 C(x, y) D2 D3 y = 1 2 x 1 2 y = 4 B =(i, j, k) E3 3. (a, b, c) R 3 ai + bj + ck =0. i BC 2 = (( BC 2 = 5 3 + 1 ) 2 ( + 4 3 3 4 ) 2 ) 1 2 0 = i (ai + bj + ck ) 3 ( 16 = 9 + 64 ) = ai i + bi j + ci k 1 2 80 = = 4 5 2, 98... 1 ( ) 9 3 3 = ai (j k)+bi (k i)+ci (i j) [i, j, k] α = ( CA, ( ) 8/3 = a CB) [2π]. CA 4/3 ( 2 u, 1) ( ) 4/3 CB b =0, c =0 8/3 ( 1 B v, 2) E3 3 B E3 u v α = u v = 4 5, i = j k [i,j,k j k =(k i) (i j). ] ( i, j ) ( u, v )= 2 1 1 2 =3> 0, j k = ( (k i) j ) i ( (k i) i ) j =[k, i, j]i =[i, j, k]i, α>0. α = 4 5 0.64... [i,j,k ]=i (j k )=i ([i, j, k]i) (j k) i D1 (1, 1), Δ C ( 5/3, 4/3 ) =[i, j, k] =[j, k, i] =[i, j, k], [i, j, k] D1 i = i j = j, k = k ( ( 1 x 5 )) ( ( +( 1) y 4 )) =0, B = B 3 3 D2 C F E A Δ D1 D3

,, E, \

E A, B, C P(E) (A\C)\(B\C) =A\(B C). (A \ C) \ (B \ C) = (A C) \ (B C) = (A C) B C = (A C) (B C) = (A C B) (A C C) = A B C = A (B C) = A \ (B C). A B B A E A, B P(E) (A \ B) (A \ C) =A \ (B C). (A \ B) (A \ C) = (A B) (A C) = A (B C) = A B C = A \ (B C).

{ y R ; x [ 1; 2], y = x 2 } =[0;4]. y R x [ 1; 2] y = x 2 x [ 1; 0] y [0 ; 1] x [0 ; 2] y [0 ; 4] y [0 ; 4] y [0 ; 4] x = yx [0 ; 2] [ 1; 2] y = x 2 P(n) n n n 0 P(n 0 ) n n n 0 P(n) P(n +1) (φ n) n N φ 0 =0, φ 1 =1 n N, φ n+2 = φ n+1 + φ n. n =0 φ 2 1 φ 2φ 0 =1 2 1 0=1=( 1) 0, n =0 n N φ 2 n+2 φ n+3 φ n+1 = φ 2 n+2 (φ n+2 + φ n+1 )φ n+1 n N, φ 2 n+1 φ n+2φ n =( 1) n. P(n) n n n 0 = (φ 2 n+2 φ n+2 φ n+1 ) φ 2 n+1 = φ n+2 (φ n+2 φ n+1 ) φ 2 n+1 = φ n+2 φ n φ 2 n+1 = (φ 2 n+1 φ n+2φ n) = ( 1) n = ( 1) n+1, n +1 n N P(n 0 ) P(n 0 +1) n n n 0 P(n) P(n +1) P(n +2)

(u n) n N u 0 =0,u 1 =1 n N, u n+2 = u n+1 + u n. 2 n N, u n > 0. n =1u 1 =1> 0 n =2 u 2 = u 1 + u 0 = 1 > 0 n =1 2 2 n =2 n n +1 n N u n > 0 u n+1 > 0 u n+1 + u n > 0, 2 n +2 n N P(n) n n n 0 P(n 0 ) n n n 0 P(n 0 ),..., P(n) P(n +1) (u n) n N u 1 =1 n N, u n+1 = u 1 + u 2 2 + + un n n n. n N, 0 <u n 1. n =1 0 <u 1 1 u 1 =1 n N k {1,..., n}, 0 <u k 1. u n+1 = u 1 + u 2 2 + + un n n n > 0+ +0 n n =0 u n+1 = u 1 + u 2 2 + + un n n n 1+ +1 n n = n n n = 1 1. nn 1 n N, 0 <u n 1.

E f : E E f f = E f f 1 = f. (x 1,x 2 ) E 2 f(x 1 )=f(x 2 ) x 1 =(f f)(x 1 )=f ( f(x 1 ) ) = f ( f(x 2 ) ) =(f f)(x 2 )=x 2. f y E y =(f f)(y) =f ( f(y) ), x E x = f(y) y = f(x) f f f f 1 f 1 = f 1 E = f 1 (f f) =(f 1 f) f = E f = f. f : E F, A P(E), A P(F ) f(a) = { y F ; a A, y = f(x) }, f 1 (A )= { x E ; f(x) A }. y F y f(a) ( a A, y = f(a) ) x E x f 1 (A ) f(x) A. E,F f : E F A P(F ) f 1( ) F (A ( ) = E f 1 (A ) ). R E x E x f 1( ) F (A ) f(x) F (A ) f(x) / A ( f(x) A ) ( x f 1 (A ) ) ( x E f 1 (A ) ), R x E, xr x ( ) R (x, y) E 2, x R y = y R x { R (x, y, z) E 3, x R y y R z = x R z.

R R (x, y) R 2 ( ), x R y x = y. R R x R x R x R, x = x x R x R x, y R x R y x = y y = x y R x, R x, y, z R { { x R y x = y = x = z x R z, y R z y = z R R R x R x R { {x, x} x 0 x = {y R ; x R y} = {y R ; x = y } = {0} x =0.

A, B E A B = B E (A). A, B E A B = A B. x R, y R, x y. y R, x R, x y. f : E F g : F G g f g f f g f : E E f f = E f f 1 = f f : E E f f = f f = E E,F f : E F A, B E f(a B) =f(a) f(b). E,F f : E F A, B E f(a B) =f(a) f(b).

E A, B, C P(E) (A B) C A (B C). A C E A, B, C P(E) A B = A C A B = A C. f R R f(x) = 3x 1 x 2. a f b f g f R\{a} R\{b} g 1 g f,g : R R x R f(x) =1+x, g(x) =x 2. f g g f. f g = g f (L n ) n N L 0 =2,L 1 =1 n N, L n+2 = L n+1 + L n. n N L 2 n+1 L n L n+2 =5( 1) n+1 n L 2 k = L n L n+1 +2 k=0 L 2n = L 2 n 2( 1) n L 2n+1 = L n L n+1 ( 1) n

R R R (x, y) R 2, ( x R y x 2 2x = y 2 2y ). R R x R x R E,F A 1,A 2 E B 1,B 2 F (A 1 B 1 ) (A 2 B 2 )=(A 1 A 2 ) (B 1 B 2 ). (A 1 B 1 ) (A 2 B 1 )=(A 1 A 2 ) B 1. (A 1 B 1 ) (A 2 B 2 )=(A 1 A 2 ) (B 1 B 2 )? E A, B, C P(E) 1) A \ B C, 2) A \ C B, 3) A B C. E,F f : E F, g : F G f g f = f f g g f g = g g f (u n ) n N u 0 =1 n N, u n Q +. n N, u n+1 = n k=0 u k k!(n k)!. E A P(E) A A : E {0, 1}, x { 0 x / A 1 x A. 1 P(E) {0, 1} 1

A, B P(E) A = B A = B, A =1 A, A B = A B, A B = A + B A B, A\B = A A B. A, B P(E) A (A B) =A A (A B) =A. E,F,G f : E F, g : F G g f f g f g g f f g E,F f : E F, g : F E g f g f g E,E f : E E A, B E A B = f(a) f(b) A f 1( f(a) ) f(a B) =f(a) f(b) f(a B) f(a) f(b) E,E f : E E A,B E A B = f 1 (A ) f 1 (B ) f ( f 1 (A ) ) f 1 (A B )=f 1 (A ) f 1 (B ) A f 1 (A B )=f 1 (A ) f 1 (B ) E A, B E A B =(A B) (A B), A B. A B E = {1, 2, 3, 4}, A = {1, 2}, B = {1, 3} E = R, A =] ;2], B =[1;+ [ (A, B) ( P(E) ) 2, A B =(A B) (B A). (A, B) ( P(E) ) 2 A B = A + B 2 A B. P(E), (A, B, C) ( P(E) ) 3, (A B) C = A (B C).

A B = A C x A B x A C B C a =2. b =3. y = f(x), x y. x R, (f g)(x) (g f)(x) x R n f : R R, x x 2 2x ( ) x x R y R, y x x R y. (a, b) (A 1 B 1 ) (A 2 B 2 ) A 1 B 2 = = B C x E f(x) x =(g ( f(x) ) x 1,x 2 E f(x 1 )=f(x 2 ) g g = g f g x 1 = x 2 n A = B a A A B x E x A, x / A x E x A B, x / A B A (A B) A (A B). (g f) g g (f g). A B y f(a) E f a A f(a) f(b) f(a B). y f(a B) E f A B x f 1 (A ) F f y f ( f 1 (A ) ) x f 1 (A B ) x f 1 (A B ) A B = {2, 3}, A B =] ;1[ ]2 ; + [. A B X, Y X =1 X, X Y = X Y, X Y = X + Y X Y.

B E (A) ( x B, x / A ) ( ( x B, x A) ) ( (A B ) ) A B =. E = {1, 2}, A= {1}, B= {2}. A B = A B. y = x +1 y x E = F = G = R, f: x x,g: y y. g f : x x = x, g f g f (x 1,x 2 ) E 2 f(x 1 ) = f(x 2 ) f ( f(x 1 ) ) = f ( f(x 2 ) ) x 1 = x 2 f y E y = f ( f(y) ) f f 1 f = f 1 E = R, f: R R, x 0 y f(a B) x A B y = f(x) x A f(x) A x B f(x) f(b) f(x) f(a) f(b) f(a B) f(a) f(b) y f(a) f(b) y f(a) y f(b) y f(a) x A y = f(x) x A B y = f(x) y f(a B) y f(b) y f(a B) f(a) f(b) f(a B) f(a B) =f(a) f(b) E = F = R, f: R R, x x 2,A= R,B= R + A B = {0}, f(a B) ={0}, f(a) =R +,f(b) =R +,f(a) f(b) =R +

(A B) C =(A C) (B C) A (B C). }{{} A (A B) C = A (B C) x A x A (B C) =(A B) C, x C. A C A C (A B) C =(A C) (B C) =A (B C). }{{} = A A C (B, C) (B, C) A B = A C = A B = A C = A B = A C = A (A B) =A (A C) = (A A) (A B) =(A A) (A C) = A B = A C. A B = A C x A B x A x B x A x A C x / A x B x / A C x A C = A C x C x A C x A B x / A x / B x A C A B A C B C A B = A C a =2. (x, y) (R \{2}) R. y = f(x) y = 3x 1 xy 2y =3x 1 x 2 xy 3x =2y 1 (y 3)x =2y 1. y 3, y = f(x) x = 2y 1 y 3 y f 2y 1 y 3. y =3, y = f(x) 0x =5, y f b =3, f g : R \{2} R \{3}, x 3x 1 x 2 f R \{2} R \{3} (x, y) (R \{2}) (R \{3}) y = g(x) y = 3x 1 x 2 x = 2y 1 y 3. y g g g g 1 : R \{3} R \{2}, y 2y 1 y 3. x R (f g)(x) =f ( g(x) ) = f(x 2 )=1+x 2 (g f)(x) =g ( f(x) ) = g(1 + x) =(1+x) 2 =1+2x + x 2. (f g)(1) = 2 (g f)(1) = 4, f g g f. n =0 L 2 n+1 L nl n+2 = L 2 1 L 0 L 2 =1 2 2 3= 5 5( 1) n+1 = 5, n =0 n N L 2 n+2 L n+1l n+3 = L 2 n+2 L n+1(l n+2 + L n+1 ) = (L 2 n+2 L n+1l n+2 ) L 2 n+1

= L n+2 (L n+2 L n+1 ) L 2 n+1 = L n+2 L n L 2 n+1 = (L 2 n+1 L nl n+2 ) = ( 5( 1) n+1) = 5( 1) n+2, n +1 n n N n n =0 L 2 k = L2 0 =22 =4, k=0 L nl n+1 +2=L 0 L 1 +2=2 1+2=4, n =0 n N n+1 ( n ) L 2 k = L 2 k + L 2 n+1 k=0 k=0 = (L nl n+1 +2)+L 2 n+1 = (L nl n+1 + L 2 n+1)+2 = L n+1 (L n + L n+1 )+2 = L n+1 L n+2 +2, n +1 n n N L 2n = L 0 =2 n =0 L 2 n 2( 1)n =2 2 2=2 L 2n+1 = L 1 =1 L nl n+1 ( 1) n =2 1 1=1, n =0 n N L 2n+2 = L 2n+1 + L 2n = ( L nl n+1 ( 1) n) + ( L 2 n 2( 1)n) = (L nl n+1 + L 2 n) 3( 1) n = L n(l n+1 + L n) 3( 1) n = L nl n+2 3( 1) n = ( L 2 n+1 5( 1) n+1) 3( 1) n = L 2 n+1 +2( 1) n = L 2 n+1 2( 1) n+1 L 2n+3 = L 2n+2 + L 2n+1 = ( L 2 n+1 2( 1)n+1) + ( L nl n+1 ( 1) n) ( ) = L n+1 Ln+1 + L n ( 1) n+1 = L n+1 L n+2 ( 1) n+1, n +1 n n N f : R R, x x 2 2x x R f(x) =f(x) x R x (x, y) R 2 x R y f(x) =f(y) f(y) =f(x) y R x (x, y, z) R 3 x R y y R z f(x) =f(y) f(y) =f(z) f(x) =f(z) x R z R R x R x x R y R y x x R y x 2 2x = y 2 2y x 2 y 2 2x +2y =0 (x y)(x + y 2) = 0 ( y = x y =2 x ). {1} x =1 x = {x, 2 x} x 1. x 1 x =1 2 x 1 (a, b) E F (a, b) (A 1 B 1 ) (A 2 B 2 ) ( ) (a, b) A 1 B 1 (a, b) A 2 B 2 ( a A 1 ) ( b B 1 a A2 ) b B 2 ( a A 1 ) ( a A 2 b B1 ) b B 2 ( a A 1 A 2 ) b B 1 B 2 (a, b) (A 1 A 2 ) (B 1 B 2 ), (A 1 B 1 ) (A 2 B 2 )=(A 1 A 2 ) (B 1 B 2 ). (a, b) E F (a, b) (A 1 B 1 ) (A 2 B 1 ) ( ) (a, b) A 1 B 1 (a, b) A 2 B 1 ( (a A 1 ) a A 2 ) b B 1 ( a A 1 A 2 ) b B 1 (a, b) (A 1 A 2 ) B 1, (A 1 B 1 ) (A 2 B 1 )=(A 1 A 2 ) B 1.

(A 1 A 2 ) (B 1 B 2 ) (a, b) a A 1 b B 2 A 1 B 1 A 2 B 2 E = F = {0, 1}, A 1 = B 1 = {0}, A 2 = B 2 = {0, 1}. (A 1 B 1 ) (A 2 B 2 )={(0, 0)} {(1, 1)} (A 1 A 2 ) (B 1 B 2 )={0, 1} {0, 1} = { (0, 0), (0, 1), (1, 0), (1, 1) }. (0, 1) = A \ B C x A x B x B C x / B x A \ B x C x B C x B C A B C = A B C x A \ B x A x / B x A A B C x B C x B C x / B x C A \ B C B C = C B (B, C) (C, B) f g f = f f x E f(x) =(f g f)(x) =f ( g f(x) ). f x =(g f)(x) =g ( f(x) ). g g f g = g g x 1,x 2 E f(x 1 )=f(x 2 ) g y 1,y 2 F x 1 = g(y 1 ) x 2 = g(y 2 ). x 1 = g(y 1 )=(g f g)(y 1 )=g (f ( g(y 1 ) )) = g ( f(x 1 ) ) = g ( f(x 2 ) ) =(g f g)(y 2 )=g(y 2 )=x 2, f u n+1 u 0,..., u n n =0u 0 =1 Q +. n N u 0,..., u n Q + n u k u n+1 = k!(n k)!, u 0,..., u n Q + k=0 0!, 1!,..., n! N u n+1 Q + n n N, u n Q +. A = B A = B A = B a A B (a) = A (a) =1, a B A B, B A A = B A = B A = B A A x E x A, x / A A (x) =1 A (x) =0 A (x) =1 A (x) x / A x A A (x) =0 A (x) =1 A (x) =1 A (x) x E, A (x) =1 A (x). A =1 A. x E x A B, x A x B A B (x) =1 A (x) =1, B (x) =1 A B (x) = A (x) B (x) ( x / A B x / A x / B A B (x) =0 A (x) =0 B (x) =0 ) A B (x) = A (x) B (x) x E, A B (x) = A (x) B (x). A B = A B A B = 1 A B = 1 A B = 1 A B = 1 (1 A )(1 B ) = 1 (1 A B + A B ) = A + B A B. A\B = A B = A B = A (1 B )= A A B. A, B P(E) A (A B) = A A B = A ( A + B A B ) = A + A B A B = A,

A (A B) =A A (A B) = A + A B A A B = A + A B A ( A B )= A + A B A B = A, A (A B) =A A A B A (A B) = A A B A A (A B) =A g f (x 1,x 2 ) E 2 f(x 1 )=f(x 2 ). g f(x 1 )=g ( f(x 1 ) ) = g ( f(x 2 ) ) = g f(x 2 ). g f x 1 = x 2. f g f z G g f x E z = g f(x). z = g ( f(x) ) f(x) F. z G, y F, z = g(y). g g f g f f g g f g g f g g f g (g f) g g = g (f g) g = g. g g 1 g f = g 1 (g f g) g 1, f f g A B y f(a). a A y = f(a) a A B a B y = f(a) f(b). f(a) f(b). a A. f(a) f(a) a f 1( f(a) ). A f 1( f(a) ). A A B = B A B f(a) f(a) f(b) f(b) f(a) f(b) = f(a) f(b) f(a B). y f(a B) x A B y = f(x) x A x B x A f(x) f(a) f(a) f(b) x B f(x) f(b) f(a) f(b) f(x) f(a) f(b) (A B) f(a) f(b) f(a B) =f(a) f(b) A B A f(a B) f(a) = A B B f(a B) f(b) = f(a B) f(a) f(b). A B x f 1 (A ) f(x) A f(x) B x f 1 (B ) f 1 (A ) f 1 (B ). y f ( f 1 (A ) ) x f 1 (A ) y = f(x) x f 1 (A )f(x) A y A f ( f 1 (A ) ) A. x E x f 1 (A B ) f(x) A B ( f(x) A f(x) B ) ( x f 1 (A ) x f 1 (B ) ) x f 1 (A ) f 1 (B ). f 1 (A B )=f 1 (A ) f 1 (B ). x E x f 1 (A B ) f(x) A B ( f(x) A f(x) B ) ( x f 1 (A ) x f 1 (B ) ) x f 1 (A ) f 1 (B ). f 1 (A B )=f 1 (A ) f 1 (B ). E = {1, 2, 3, 4}, A= {1, 2}, B= {1, 3}, A B = {1, 2, 3}, A B = {1}, A B = {2, 3, 4}, A B = {2, 3}.

E = R, A =] ;2], B =[1;+ [, A B = R, A B =[1;2], A B =] ;1[ ]2 ; + [, A B =] ;1[ ]2 ; + [. (A, B) ( P(E) ) 2 A B =(A B) (A B) =(A B) (A B) =(A A) (A B) (B A) (B B) =(A B) (B A). (A, B) ( P(E) ) 2 A B = (A B) (B A) = A B + B A A B B A }{{} =0 = A (1 B )+ B (1 A )= A + B 2 A B. (A, B, C) ( P(E) ) 3. (A B) C = A B + C 2 A B C =( A + B 2 A B )+ C 2 ( A + B 2 A B ) C = A + B + C 2( A B + A C + B C )+4 A B C. A (B C) = A + B C 2 A B C = A +( B + C 2 B C ) 2 A ( B + C 2 B C ) = A + B + C 2( A B + A C + B C )+4 A B C. (A B) C = A (B C). (A B) C = A (B C), P(E).

n n n k, k 2, q k k=1 k=1 k=0 a n b n n N ( ) n, p ( ) n n! = p p!(n p)! ( ) ( ) ( ) n n n +1 + = p p +1 p +1

n N, q R \{1}, n q=0 q k = 1 qn+1 1 q n k = k=1 n(n +1), 2 n k 2 = k=1 n(n + 1)(2n +1) 6 n N, (x, y) R 2, (x + y) n = n k=0 ( ) n x k y n k. k n N n ( 1) k (2k +1)=( 1) n (n +1). k=0 n n =0 n N n ( 1) k (2k +1)=( 1) n (n +1). k=1 n+1 n ( 1) k (2k +1) = ( 1) k (2k +1)+( 1) n+1 (2n +3) k=0 k=0 = ( 1) n (n +1)+( 1) n+1 (2n +3) = ( 1) n+1( (n +1)+(2n +3) ) = ( 1) n+1 (n +2), n +1 n N