當前位置:編程學習大全網 - 編程語言 - 對於無窮數列{xn}和函數f(x),若xn+1=f(xn)(n∈N+),則稱f(x)是數列{xn}的母函數.(Ⅰ)定義在R

對於無窮數列{xn}和函數f(x),若xn+1=f(xn)(n∈N+),則稱f(x)是數列{xn}的母函數.(Ⅰ)定義在R

(I)(1)由題知a1=g(
1
2
)=1,

an+1=g(

1
2n+1
)=g(
1
2
?
1
2n
)=
1
2
g(
1
2n
)+
1
2n
g(
1
2
)=
1
2
g(
1
2n
)+
1
2n

an+1=
1
2
an+
1
2n
?2n+1an+1=2nan+2,

∴f(x)=x+2是數列{2nan}的母函數.

(2)由(1)可知:數列{2nan}是等差數列,首項為2a1=2,公差d=2,

∴2nan=2+(n-1)×2=2n,解得an=

n
2n?1

∴Sn=1+

2
2
+
3
22
+…+
n
2n?1

1
2
Sn=
1
2
+
2
22
+…+
n?1
2n?1
+
n
2n

1
2
Sn=1+
1
2
+
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