If $f: R \to R$ is a non constant, thrice differentiable function such that $f(1+ \frac{1}{n}) = 1$ for all integers, then find $f''(1)$

Question

If $f: R \to R$ is a non constant, thrice differentiable function such that $f(1+ \frac{1}{n}) = 1$ for all integers, then find $f''(1)$

Since function is continuous so, $f(1) = 1$

I have this formula for $f''$

$f''(c) =\lim_{h \to 0} \frac{f(c+h) + f(c-h) - 2f(c)}{h^{2}}$ So $ f''(1) =\lim_{n \to \infty}\frac{f(1 + 1/n) + f(1- 1/n) - 2f(1)}{1/n^{2}}$ Then I used L' hospital, but ended up getting $f''(1+1/n)$ which is just $f''(1)$

Any suggestion or hint would be helpful.