An odd function is symmetric about the vertical line `x=a ,(a >0),a n dif` `sum_(r=0)^oo[f(1+4r]^r=8,` then find the value of `f(1)dot`

Explanation

f(x) is an odd function. Therefore,
`f(x)= -f(-x) " (1)" `
`f(x)` is symmetrical about the line`x=a`. Therefore,
`f(a-x)=f(a+x) " (2)" `
` :. f(2a-x)=f(x) " "("Replacing " x " by " a-x)`
or `f(2a+x)=f(-x) " "("Replacing " x " by " -x)`
or `f(2a+x)= -f(x) " " ( :' f " is odd") `
or `f(x+4a)= -f(x+2a) " " ("Replacing " x " by " x+2a)`
or `f(x+4a)= f(x)`
i.e., `f` is periodic with period `4a`. So,
or `f(1+4r)= f(1)`
Now `sum_(r=0)^(oo)[f(1)]^(r)=8`
or `(1)/(1-f(1))=8`
or `f(1)=7//8`