Question

¹ýµãP£¨1£¬0£©×÷ÇúÏßC£ºy=x^k£¨x¡Ê£¨0£¬+¡Þ£©£¬k¡ÊN^*£¬k£¾1£©µÄÇÐÏߣ¬ÇеãΪM₁£¬ÉèM₁ÔÚxÖáÉϵÄͶӰÊǵãP₁£»ÓÖ¹ýµãP₁×÷ÇúÏßCµÄÇÐÏߣ¬ÇеãΪM₂£¬ÉèM₂ÔÚxÖáÉϵÄͶӰÊǵãP₂£»¡­£»ÒÀ´ËÏÂÈ¥£¬µÃµ½Ò»ÏµÁеãM₁£¬M₂£¬¡­M_n£¬¡­£»ÉèËüÃǵĺá×ø±êa₁£¬a₂£¬¡­£¬
a_n¡­¹¹³ÉÊýÁÐΪ{a_n}£®
£¨¢ñ£©ÇóÊýÁÐ{a_n}µÄͨÏʽ£»
£¨¢ò£©ÇóÖ¤£ºan¡Ý1+

n

k-1

£»
£¨¢ó£©µ±k=2ʱ£¬Áîbn=

n

an

£¬ÇóÊýÁÐ{b_n}µÄÇ°nÏîºÍS_n£®

Answer 1

£¨¢ñ£©¶Ôy=x^kÇóµ¼Êý£¬
µÃy¡ä=kx^k-1£¬
µãÊÇM_n£¨a_n£¬a_n^k£©µÄÇÐÏß·½³ÌÊÇy-a_n^k=ka_n^k-1£¨x-a_n£©£®¡­£¨2·Ö£©
µ±n=1ʱ£¬ÇÐÏß¹ýµãP£¨1£¬0£©£¬
¼´0-a₁^k=ka₁^k-1£¨1-a₁£©£¬
µÃa1=

k

k-1

£»
µ±n£¾1ʱ£¬ÇÐÏß¹ýµãP_n-1£¨a_n-1£¬0£©£¬
¼´0-a_n^k=ka_n^k-1£¨a_n-1-a_n£©£¬
µÃ

an

an-1

=

k

k-1

£®
ËùÒÔÊýÁÐ{a_n}ÊÇÊ×Ïîa1=

k

k-1

£¬¹«±ÈΪ

k

k-1

µÄµÈ±ÈÊýÁУ¬
ËùÒÔÊýÁÐ{a_n}µÄͨÏʽΪan=(

k

k-1

)n£¬n¡ÊN*£®¡­£¨4·Ö£©
£¨ II£©Ó¦ÓöþÏîʽ¶¨Àí£¬µÃan=(

k

k-1

)n=(1+

1

k-1

)n=

C

0n

+

C

1n

1

k-1

+

C

2n

(

1

k-1

)2+¡­+

C

nn

(

1

k-1

)n¡Ý1+

n

k-1

£®¡­£¨8·Ö£©
£¨ III£©µ±k=2ʱ£¬a_n=2ⁿ£¬
ÊýÁÐ{b_n}µÄÇ°nÏîºÍS_n=

1

2

+

2

22

+

3

23

+¡­+

n

2n

£¬
ͬ³ËÒÔ

1

2

£¬µÃ

1

2

Sn=

1

22

+

2

23

+

3

24

+¡­+

n

2n+1

£¬
Á½Ê½Ïà¼õ£¬¡­£¨10·Ö£©
µÃ

1

2

Sn=

1

2

+

1

22

+

1

23

+¡­+

1

2n

-

n

2n+1

=

1 2 (1- 1 2n )

1- 1 2

-

n

2n+1

=1-

1

2n

-

n

2n+1

£¬
ËùÒÔS_n=2-

n+2

2n

£®¡­£¨12·Ö£©