Let the function such that
f(x-1)+f(x+1)=31/2f(x), x€R.If f(5)=100, find

_r=0⁹⁹∑f(5+12r).

Hello Student,

f(x-1)+f(x+1)=3^1/2f(x)
=>f(x)+f(x+2)=3^1/2f(x+1)
x to x+2
f(x+1)+f(x+3)=3^1/2f(x+2)
=>f(x-1)+2f(x+2)+f(x+3)=sqrt3(sqrt3(f(x+1)))
=>f(x-1)+f(x+3)=f(x+1)
x to x+2
f(x+1)+f(x+5)=f(x+3)
f(x-1)+f(x+5)=0
=>f(x)=-f(x+6)
=>f(x)=f(x+12)
=> period =12
=>sum =100*f(5)

Thanks & Regards
Arun Kumar
Btech, IIT Delhi
Askiitians Faculty