设f(x)=1+lnx2-x,则f(12013)+f(22013)+f(32013)+…+f(40252013)=______.-数学

题目简介

设f(x)=1+lnx2-x,则f(12013)+f(22013)+f(32013)+…+f(40252013)=______.-数学

题目详情

f(x)=1+ln
x
2-x
,则f(
1
2013
)+f(
2
2013
)+f(
3
2013
)+…+f(
4025
2013
)
=______.
题型:填空题难度:中档来源:不详

答案

f(x)=1+lnclass="stub"x
2-x

∴f(t)+f(2-t)=1+lnclass="stub"t
2-t
+1+lnclass="stub"2-t
t
=2+lnclass="stub"t
2-t
•class="stub"2-t
t
=2
f(class="stub"1
2013
)+f(class="stub"4025
2013
)
=2,f(class="stub"2
2013
)+f(class="stub"4024
2013
)=…=f(class="stub"4025
2013
)+f(class="stub"1
2013
)
=2
∴S=f(class="stub"1
2013
)+f(class="stub"2
2013
)+f(class="stub"3
2013
)+…+f(class="stub"4025
2013
)

S=f(class="stub"4025
2013
)+f(class="stub"4024
2013
)
…+f(class="stub"1
2013
)

2S=2×4025
∴S=4025
故答案为:4025

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