if f(x)=9 logical and x divided by left parenthesis 9 logical and x plus 3 right parenthesis then what is the value of f(1/1996)+f(2/1996)...+f(1995/1996)

Asked by Devesh | 1st Nov, 2015, 05:46: PM

Expert Answer:

G i v e n space t h a t f open parentheses x close parentheses equals fraction numerator 9 to the power of x over denominator 3 plus 9 to the power of x end fraction... left parenthesis 1 right parenthesis L e t space u s space c o n s i d e r space f open parentheses 1 minus x close parentheses colon f open parentheses 1 minus x close parentheses equals fraction numerator 9 to the power of 1 minus x end exponent over denominator 3 plus 9 to the power of 1 minus x end exponent end fraction... left parenthesis 2 right parenthesis N o w space a d d i n g space e q u a t i o n s space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space w e space h a v e comma f open parentheses x close parentheses plus f open parentheses 1 minus x close parentheses equals fraction numerator 9 to the power of x over denominator 3 plus 9 to the power of x end fraction plus fraction numerator 9 to the power of 1 minus x end exponent over denominator 3 plus 9 to the power of 1 minus x end exponent end fraction space space space space space space space space space space space space space space space space space space space space space equals fraction numerator open parentheses 3 plus 9 to the power of 1 minus x end exponent close parentheses cross times 9 to the power of x plus 9 to the power of 1 minus x end exponent cross times open parentheses 3 plus 9 to the power of x close parentheses over denominator open parentheses 3 plus 9 to the power of x close parentheses open parentheses 3 plus 9 to the power of 1 minus x end exponent close parentheses end fraction space space space space space space space space space space space space space space space space space space space space space equals fraction numerator open parentheses 3 cross times 9 to the power of x plus 9 to the power of 1 close parentheses plus open parentheses 3 cross times 9 to the power of 1 minus x end exponent plus 9 to the power of 1 close parentheses over denominator open parentheses 3 plus 9 to the power of x close parentheses open parentheses 3 plus 9 to the power of 1 minus x end exponent close parentheses end fraction space space space space space space space space space space space space space space space space space space space space space equals fraction numerator open parentheses 3 plus 9 to the power of x close parentheses open parentheses 3 plus 9 to the power of 1 minus x end exponent close parentheses over denominator open parentheses 3 plus 9 to the power of x close parentheses open parentheses 3 plus 9 to the power of 1 minus x end exponent close parentheses end fraction therefore f open parentheses x close parentheses plus f open parentheses 1 minus x close parentheses equals 1... left parenthesis 3 right parenthesis N o w space s u b s t i t u t i n g space x equals 1 over 1996 space i n space e q u a t i o n space left parenthesis 3 right parenthesis comma space w e space g e t comma f open parentheses 1 over 1996 close parentheses plus f open parentheses 1 minus 1 over 1996 close parentheses equals 1 rightwards double arrow f open parentheses 1 over 1996 close parentheses plus f open parentheses 1995 over 1996 close parentheses equals 1... left parenthesis 4 right parenthesis S i m i l a r l y comma space w h i l e space s u b s t i t u t i n g space x equals 2 over 1996 comma space 3 over 1996 comma 4 over 1996 comma 5 over 1996 comma.... comma 997 over 1996 space i n space e q u a t i o n space left parenthesis 3 right parenthesis comma space w e space g e t f open parentheses 2 over 1996 close parentheses plus f open parentheses 1994 over 1996 close parentheses equals 1... left parenthesis 5 right parenthesis f open parentheses 3 over 1996 close parentheses plus f open parentheses 1993 over 1996 close parentheses equals 1... left parenthesis 6 right parenthesis f open parentheses 4 over 1996 close parentheses plus f open parentheses 1992 over 1996 close parentheses equals 1... left parenthesis 7 right parenthesis f open parentheses 5 over 1996 close parentheses plus f open parentheses 1991 over 1996 close parentheses equals 1... left parenthesis 8 right parenthesis . . . f open parentheses 997 over 1996 close parentheses plus f open parentheses 999 over 1996 close parentheses equals 1... left parenthesis 1000 right parenthesis C o n s i d e r space t h e space e x p r e s s i o n comma f open parentheses 1 over 1996 close parentheses plus f open parentheses 2 over 1996 close parentheses plus f open parentheses 3 over 1996 close parentheses plus f open parentheses 4 over 1996 close parentheses plus f open parentheses 5 over 1996 close parentheses plus space space space space space space space... plus f open parentheses 997 over 1996 close parentheses plus f open parentheses 998 over 1996 close parentheses plus f open parentheses 999 over 1996 close parentheses plus... space space space space space space space space... f open parentheses 1991 over 1996 close parentheses plus f open parentheses 1992 over 1996 close parentheses plus f open parentheses 1993 over 1996 close parentheses plus f open parentheses 1994 over 1996 close parentheses plus f open parentheses 1995 over 1996 close parentheses colon R e w r i t i n g space t h e space a b o v e space e x p r e s s i o n comma space w e space h a v e comma f open parentheses 1 over 1996 close parentheses plus f open parentheses 2 over 1996 close parentheses plus f open parentheses 3 over 1996 close parentheses plus f open parentheses 4 over 1996 close parentheses plus f open parentheses 5 over 1996 close parentheses plus space space space space space space space... plus f open parentheses 997 over 1996 close parentheses plus f open parentheses 998 over 1996 close parentheses plus f open parentheses 999 over 1996 close parentheses plus... space space space space space space space space... f open parentheses 1991 over 1996 close parentheses plus f open parentheses 1992 over 1996 close parentheses plus f open parentheses 1993 over 1996 close parentheses plus f open parentheses 1994 over 1996 close parentheses plus f open parentheses 1995 over 1996 close parentheses equals f open parentheses 1 over 1996 close parentheses plus f open parentheses 1995 over 1996 close parentheses plus f open parentheses 2 over 1996 close parentheses plus f open parentheses 1994 over 1996 close parentheses plus f open parentheses 3 over 1996 close parentheses plus f open parentheses 1993 over 1996 close parentheses plus f open parentheses 4 over 1996 close parentheses space space space space space plus f open parentheses 1992 over 1996 close parentheses plus f open parentheses 5 over 1996 close parentheses plus f open parentheses 1991 over 1996 close parentheses plus... plus f open parentheses 997 over 1996 close parentheses plus f open parentheses 999 over 1996 close parentheses plus f open parentheses 998 over 1996 close parentheses H e n c e space w e space n e e d space t o space f i n d space t h e space v a l u e space o f space t h e space a b o v e space e x p r e s s i o n. T h u s comma space a d d i n g space t h e space e q u a t i o n s space left parenthesis 4 right parenthesis comma space left parenthesis 5 right parenthesis comma space left parenthesis 6 right parenthesis comma space left parenthesis 7 right parenthesis comma space left parenthesis 8 right parenthesis comma space... left parenthesis 1000 right parenthesis comma space w e space h a v e comma f open parentheses 1 over 1996 close parentheses plus f open parentheses 1995 over 1996 close parentheses plus f open parentheses 2 over 1996 close parentheses plus f open parentheses 1994 over 1996 close parentheses plus f open parentheses 3 over 1996 close parentheses plus f open parentheses 1993 over 1996 close parentheses plus f open parentheses 4 over 1996 close parentheses space space space space space plus f open parentheses 1992 over 1996 close parentheses plus f open parentheses 5 over 1996 close parentheses plus f open parentheses 1991 over 1996 close parentheses plus... plus f open parentheses 997 over 1996 close parentheses plus f open parentheses 999 over 1996 close parentheses plus f open parentheses 998 over 1996 close parentheses equals stack stack open parentheses 1 plus 1 plus 1 plus 1 plus 1... plus 1 close parentheses with underbrace below with 997 space t e r m s below plus f open parentheses 1 half close parentheses rightwards double arrow f open parentheses 1 over 1996 close parentheses plus f open parentheses 1995 over 1996 close parentheses plus f open parentheses 2 over 1996 close parentheses plus f open parentheses 1994 over 1996 close parentheses plus f open parentheses 3 over 1996 close parentheses plus f open parentheses 1993 over 1996 close parentheses plus f open parentheses 4 over 1996 close parentheses space space space space space plus f open parentheses 1992 over 1996 close parentheses plus f open parentheses 5 over 1996 close parentheses plus f open parentheses 1991 over 1996 close parentheses plus... plus f open parentheses 997 over 1996 close parentheses plus f open parentheses 999 over 1996 close parentheses plus f open parentheses 998 over 1996 close parentheses equals 997 plus 0.5 equals 997.5 space space space space open square brackets because f open parentheses 1 half close parentheses equals 0.5 close square brackets

Answered by Vimala Ramamurthy | 2nd Nov, 2015, 10:10: AM