f ( x ) = 4 x 4 x + 2 {\displaystyle f(x)={\frac {4^{x)){4^{x}+2))} ,求 f ( 1 79 ) + f ( 2 79 ) + f ( 3 79 ) + ⋯ ⋯ + f ( 77 79 ) + f ( 78 79 ) = {\displaystyle f({\frac {1}{79)))+f({\frac {2}{79)))+f({\frac {3}{79)))+\cdots \cdots +f({\frac {77}{79)))+f({\frac {78}{79)))=} ?
f ( x ) = 4 x 4 x + 2 = 1 − 2 4 x + 2 {\displaystyle {\displaystyle f(x)={\frac {4^{x)){4^{x}+2))=1-{\frac {2}{4^{x}+2)))) ∑ i = 1 78 f ( i 79 ) {\displaystyle {\displaystyle \sum _{i=1}^{78}f({\frac {i}{79))))) = ∑ i = 1 78 ( 1 − 2 4 i 79 + 2 ) {\displaystyle {\displaystyle =\sum _{i=1}^{78}(1-{\frac {2}{4^{\frac {i}{79))+2))))) = 78 − ∑ i = 1 78 2 4 i 79 + 2 {\displaystyle {\displaystyle =78-\sum _{i=1}^{78}{\frac {2}{4^{\frac {i}{79))+2)))) = 78 − ∑ i = 1 39 ( 2 4 i 79 + 2 + 2 4 79 − i 79 + 2 ) {\displaystyle {\displaystyle =78-\sum _{i=1}^{39}({\frac {2}{4^{\frac {i}{79))+2))+{\frac {2}{4^{\frac {79-i}{79))+2))))) = 78 − 2 ∑ i = 1 39 4 79 − i 79 + 4 i 79 + 4 ( 4 i 79 + 2 ) ( 4 79 − i 79 + 2 ) {\displaystyle {\displaystyle =78-2\sum _{i=1}^{39}{\frac {4^{\frac {79-i}{79))+4^{\frac {i}{79))+4}{(4^{\frac {i}{79))+2)(4^{\frac {79-i}{79))+2))))) = 78 − 2 ∑ i = 1 39 4 79 − i 79 + 4 i 79 + 4 4 + 2 ∗ 4 79 − i 79 + 2 ∗ 4 i 79 + 4 {\displaystyle {\displaystyle =78-2\sum _{i=1}^{39}{\frac {4^{\frac {79-i}{79))+4^{\frac {i}{79))+4}{4+2*4^{\frac {79-i}{79))+2*4^{\frac {i}{79))+4)))) = 78 − ∑ i = 1 39 4 79 − i 79 + 4 i 79 + 4 4 79 − i 79 + 4 i 79 + 4 {\displaystyle {\displaystyle =78-\sum _{i=1}^{39}{\frac {4^{\frac {79-i}{79))+4^{\frac {i}{79))+4}{4^{\frac {79-i}{79))+4^{\frac {i}{79))+4)))) = 78 − 39 = 39 {\displaystyle {\displaystyle =78-39=39)) 😵😵
f ( 1 79 ) + f ( 78 79 ) {\displaystyle f({\cfrac {1}{79)))+f({\cfrac {78}{79)))} = 4 1 79 4 1 79 + 2 + 4 78 79 4 78 79 + 2 {\displaystyle ={\cfrac {4^{\frac {1}{79))}{4^{\frac {1}{79))+2))+{\cfrac {4^{\frac {78}{79))}{4^{\frac {78}{79))+2))} = ( 4 1 79 ) ( 4 78 79 + 2 ) + ( 4 78 79 ) ( 4 1 79 + 2 ) ( 4 1 79 + 2 ) ( 4 78 79 + 2 ) {\displaystyle ={\cfrac {(4^{\frac {1}{79)))(4^{\frac {78}{79))+2)+(4^{\frac {78}{79)))(4^{\frac {1}{79))+2)}(((4^{\frac {1}{79))+2)}{(4^{\frac {78}{79))+2))))) = ( 4 + 2 × 4 1 79 ) + ( 4 + 2 × 4 78 79 ) 8 + 2 ( 4 1 79 + 4 78 79 ) = 1 {\displaystyle ={\cfrac {(4+2\times 4^{\frac {1}{79)))+(4+2\times 4^{\frac {78}{79)))}{8+2(4^{\frac {1}{79))+4^{\frac {78}{79)))))=1} 同理 f ( 2 79 ) + f ( 77 79 ) = 1 {\displaystyle f({\cfrac {2}{79)))+f({\cfrac {77}{79)))=1} , f ( 3 79 ) + f ( 76 79 ) = 1 {\displaystyle f({\cfrac {3}{79)))+f({\cfrac {76}{79)))=1} ……, 所以 f ( 1 79 ) + f ( 2 79 ) + . . . + f ( 78 79 ) {\displaystyle f({\cfrac {1}{79)))+f({\cfrac {2}{79)))+...+f({\cfrac {78}{79)))} = [ f ( 1 79 ) + f ( 78 79 ) ] + [ f ( 2 79 ) + f ( 77 79 ) ] + . . . + [ f ( 39 79 ) + f ( 40 79 ) ] {\displaystyle =[f({\cfrac {1}{79)))+f({\cfrac {78}{79)))]+[f({\cfrac {2}{79)))+f({\cfrac {77}{79)))]+...+[f({\cfrac {39}{79)))+f({\cfrac {40}{79)))]} = 39 × 1 = 39 {\displaystyle =39\times 1=39}