+ + + + n S (n) = + + + + n S (n) S (n) S 0 (n) S (n) 6 4 S (n) S (n) 7 S (n) S 4 (n) 8 6 S k (n) 0 7 (k + )S k (n) 8 S 6 (n), S 7 (n), S 8 (n), S 9 (

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k k + k + k + + n k 006.7.

+ + + + n S (n) = + + + + n S (n) S (n) S 0 (n) S (n) 6 4 S (n) S (n) 7 S (n) S 4 (n) 8 6 S k (n) 0 7 (k + )S k (n) 8 S 6 (n), S 7 (n), S 8 (n), S 9 (n), S 0 (n) 9 S (n) S 4 (n) 6 0 S k (n) 7 0

+ + + + n k + k + k + + n k n n(n + )(n + ) 6 (k + ) = k + k + k + (k + ) k = k + k + n =,,, = + + = + + 4 = + +... (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 + ) n(n + ) (n + ) = (n + ){(n + ) n } = (n + ){n + 4n + n } = (n + ){n + n} n(n + )(n + )} =.

+ + + + n = n(n + )(n + )} 6 4

S (n) = + + + + n + + + + = n = n n(n + ) + + + + n = = n + + + + + n n(n + )(n + ) = = 6 n + n + 6 n 0 + 0 + 0 + + n 0 = n () + + + + n = n + n () + + + + n = n + n + 6 n () 0 + 0 + 0 + + n 0 = S 0 (n) + + + + n = S (n) + + + + n = S (n). S 0 (n) = n S (n) = n + n S (n) = n + n + 6 n S (n) S 4 (n) S (n) S (n) S 4 (n)

S (n) S (n) S 0 (n) S (n) S (n) S (n) S (n) S (n) S (n) S (n) S (n) S(n) 0 S 0 (n) = ( n + n + 6 n)0 = n + n + 6 6 S (n) = n + n S (n) S (n) S 0 (n) S (n) S (n) S 0 (n) = ( n + n)0 = n + S 0(n) = n S (n) S 0(n) S 0 (n) S (n) S 0 (n) S (n) S (0) = 0 = 0 S () = = S (n) S (n) S (n) S (n) S (0) = 0 = 0 S () = = 6

4 S (n) S (n) S (n) + + + S (n) = 7

S (n) S 4 (n) S (n) = 4 n4 + n + 4 n S (n) S 4 (n) S (n) 4 S 4 (n) S 4 (0) = 0 4 = 0 S 4 () = 4 = S (n) = 4 n4 + n + 4 n Z 4S (n) = n 4 + n + n 4S (n) + C = n 4 + n + n + C Z (4S (n) + C)dn = (n 4 + n + n + C)dn = n + n4 + n + Cn + C S 4 (n) = n + n4 + n + Cn + C S 4 (0) = 0 4 = 0 S 4 () = 4 = C = 0 + + + C = 6 + + 0 + C 0 = 0 + C = C = 0 = 0 S 4 (n) = n + n4 + n 0 n 8

n = S 4 (n) = n + n4 + n 0 n S 4 () = + 4 + 0 = + 6 + 8 0 = + 8 + 8 = = = 7 96 + 0 + 40 4 + 4 = + 6 = 7 n = n = S 4 (n) = n + n4 + n 0 n S 4 () = + 4 + 0 = 4 + 8 + 7 0 = 4 + 8 + 9 0 = = 980 0 = 98 486 + 40 + 90 0 4 + 4 + 4 = + 6 + 8 = 98 4 + 4 + 4 + + n 4 = n + n4 + n 0 n 9

6 S k (n) S 0 (n) = + + + + = n S (n) = + + + + n = n + n S (n) = + + + + n = n + n + 6 n S (n) = + + + + n = 4 n4 + n + 4 n S 4 (n) = 4 + 4 + 4 + + n 4 = n + n4 + n 0 n S 0 (n) = + + + + = n S (n) = + + + + n = n + n S (n) = + + + + n = n + n + 6 n S (n) = + + + + n = 4 n4 + n + 4 n + 0n S 4 (n) = 4 + 4 + 4 + + n 4 = n + n4 + n + 0n 0 n S (n) S 4 (n) S 4 (n) = n + n4 + n + 0n 0 n S (n) = an 6 + bn + cn 4 + dn + en + fn S 4 (n) S (n) e a = 6 b = c = 4 = = = 6 d = 0 e = 0 = 0 = 0

f S () = a + b + c + d + e + f = 6 + + + f = + 6 + + f = + f = f = 0 S 4 (n), 4S (n), S (n), S 0 (n) = n S (n) = n + n S (n) = n + n + n 4S (n) = n 4 + n + n S 4 (n) = n + n4 + n 6 n 6S (n) = n 6 + n + n4 n

7 (k + )S k (n) S 0 (n) S (n) S (n) 4S (n) 0 S 4 (n) 6S (n) n 0 6 0 y. y y. y 4 4 y 4 y 4. 4 6 0 0 0 y y y 4 y. 0 6 y y 6 6 y 6 y 4 6 y 6 6. 6 0 0 0

S 0 (n) y. S (n) y y S (n) y 4 4 y 4. y 4 6. 4 4S (n) 0 0 y y y y. 4 S 4 (n) 0 6 y 6 y 6 y 6 y 6 y 6 4 6 6S (n) 0 0 n 6 n n 4 n n n 0. 6 0 6S (n) n 6 S 0 (n) n 4 6 4 6 = 4 6 4 6 = 6 C 6 = 6 C 0 6S (n) n S (n) n 4 6 4 = 4 6 4 = 6 C = 6 C 6S (n) n 4 S (n) n 6 4 6 4 = 4 6 4 = 6 C 4 = 6 C 6S (n) n S (n) n 0 4 6 = 4 6 = 6 C 6S (n) n S 4 (n) n 0 6 = 6 = 6 C = 6 C 4 6S (n) n S (n) n 0 6 = 6 = 6 C = 6 C

S 0 (n) n B S (n) n B S (n) n 6 B S (n) n 0 B 4 S 4 (n) n 0 B S (n) n 0 B 6 6S (n) n 6 6 C 0 B 6S (n) n 6 C B 6S (n) n 4 6 C B 6S (n) n 6 C B 4 6S (n) n 6 C 4 B 6S (n) n 6 C B 6 6S (n) = 6 C 0 B n 6 + 6 C B n + 6 C B n 4 + 6 C B 4 n + 6 C 4 B n + 6 C B 6 n () 6S (n) n B 6 S 4 (n) n = S ( ) = = 6S () = 6 C 0 B + 6 C B + 6 C B + 6 C B 4 + 6 C 4 B + 6 C B 6 = 6 6 6 C 0 B 6 C B 6 C B 6 C B 4 6 C 4 B B 6 = 6 B 0 =, B =, B = 6, B = 0, B 4 = 0, B = 0, Ars Conjectandi 4

8 S 6 (n), S 7 (n), S 8 (n), S 9 (n), S 0 (n) S 6 (n), S 7 (n), S 8 (n), S 9 (n), S 0(n)

9 S (n) S 4 (n) S (n) S 4 (n) S 4 (n) S (n) S k (n) S k+ (n) S (n) = + + + + (n ) + n S (n ) = + + + + (n ) S 4 (n) = 4 + 4 + 4 + + (n ) 4 + n 4 ) S 4 (n ) = 4 + 4 + 4 + + (n ) 4 S 4 (n) S 4 (n ) = n 4 S 0 4(n) S 0 4(n ) = 4 n S4(n) 0 S4(n 0 ) = 4 n S4(n 0 ) S4(n 0 ) = 4 (n ). S4() 0 S4() 0 = 4 S4() 0 S4() 0 = 4 S4() 0 S4(0) 0 = 4 S4(n) 0 S4(0) 0 = 4{ + + + n } = 4S (n) S 0 4(n) c = 4S (n) Z S4(n) 0 = 4S (n) + c Z S4(n)dn 0 = {4S (n) + c}dn Z S 4 (n) = {4S (n) + c}dn S (n) S 4 (n) S (n) c S 4 (n) c n = 0, S 4 (0) = 0, S 4 () = 6

0 S k (n) 7 S (n) 6S (n) = 6 C 0 B n 6 + 6 C B n + 6 C B n 4 + 6 C B 4 n + 6 C 4 B n + 6 C B 6 n () 6 C 0 6 C 6 B 0 B B 6 n = S () = 6 = 6 C 0 B + 6 C B + 6 C B + 6 C B 4 + 6 C 4 B + 6 C B 6 B 6 S k (n) 4 6 4 0 0 6 0 6 7 7 B k B 0 B 0 B B 0 B B B 0 B B B B 0 4B 6B 4B B 4 B 0 B 0B 0B B 4 B B 0 6B B 0B B 4 6B B 6 B 0 7B B B B 4 B 7B 6 B 7 B k k 7

0 B 0 B 0 B B 0 B B B 0 4B 6B 4B 4 B 0 B 0B 0B B 4 B 0 6B B 0B B 4 6B 6 B 0 7B B B B 4 B 7B 6 7 0 B 0 = B 0 + B = B 0 + B + B = B 0 + 4B + 6B + 4B = 4 B 0 + B + 0B + 0B + B 4 = B 0 + 6B + B + 0B + B 4 + 6B = 6 B 0 + 7B + B + B + B 4 + B + 7B 6 = 7 B 0 =, B =, B = 6, B = 0, B 4 = 0, B = 0, B 6 = 4, B 7 = 0, B 8 = 0, S k (n) 0 B 0 n = S 0 (n) B 0 n + B n = S (n) B 0 n + B n + B n = S (n) B 0 n 4 + 4B n + 6B n + 4B n = 4S (n) B 0 n + B n 4 + 0B n + 0B n + B 4 n = S 4 (n) B 0 n 6 + 6B n + B n 4 + 0B n + B 4 n + 6B n = 6S (n) B 0 n 7 + 7B n 6 + B n + B n 4 + B 4 n + B n + 7B 6 n = 7S 6 (n) B 0 =, B =, B = 6, B = 0, B 4 = 0, B = 0, B 6 = 4, B 7 = 0, B 8 = 0, 8

0 n = S 0 (n) n + n = S (n) n + n + 6 n = S (n) n 4 + 4 n + 6 6 n + 0 = 4S (n) n + n4 + 0 6 n + 0 + 0 n = S4 (n) n 6 + 6 n + 6 n4 + 0 + n + 0 = 6S (n) 0 n 7 + 7 n6 + 6 n + 0 + n + 0 + 7 0 4 n = 7S 6(n) 0 n = S 0 (n) n + n = S (n) n + n + 6 n = S (n) 4 n4 + n + 4 n + 0 = S (n) n + n4 + n + 0 0 n = S 4(n) 6 n6 + n + n4 + 0 n + 0 = S (n) 7 n7 + n6 + n + 0 6 n + 0 + 4 n = S 6(n) S k (n) 9

6 S k (n) 4 6 7 8 9 0 4 9 6 6 49 64 8 00 8 7 64 6 4 79 000 6 8 6 6 96 40 4096 66 0000 4 04 7776 6807 768 9049 00000 6 0 8 6 4 4 0 9 40 04 8 8 9 6 00 44 784 96 0 0 7 98 4 979 7 4676 877 76 00 44 0 9008 6776 08 08 x y = f(x) = a n x n + a n x n + + a x x f(x ) z = f(x ) x δy = z z x x δ y = δy δy x x δ y = δ y δ y x 4 x x f(x ) z = f(x ) x δy = z z x x δ y = δy δy x 4 x δ y = δ y δ y x x x f(x ) z = f(x ) δy = z 4 z δ y = δy 4 δy δ y = δ y 4 δ y x x 4 x x x x 6 x x 4 f(x 4 ) z 4 = f(x 4) δy 4 = z z 4 δ y 4 = δy δy 4 δ y 4 = δ y δ y 4 x 4 x x 4 x 6 x 4 x 7 x 4 z = z = z = a δy = δy = δy = a n δ n y = δ n y = δ n y = n a n f(x) ax n n n 0