Summation of infinte series

Question

Sir, I have three infinite summation

$A =J_1 \sum_{n=2}^\infty (n-1) f(n-2) \tag 1$ , $B =\sum_{n=0}^\infty f(n) \tag 2$ and $C =J_2\sum_{n=1}^\infty f(n-1) \tag 3$, with $f(0)=2,f(1)=5$ , $J_1$ and $J_2$ are constants

Question

Can we express $A $ as functions of $B $ and $C $ if possible? Means can we rewrite $A $ using $B $ and $C $ only

Note

Hint is that $\sum_{n=0}^\infty f(n) $ is a constant called $\psi$ but we are not aware of the value of it. It implies derivative of $\sum_{n=0}^\infty f(n) $ is $0$ only . Thanks

Answer 1

If you write $C$ in full, it is $J_2(f(0)+f(1)+f(2)+...)$, which is the same as $J_2B$.

If you write $A$ in full, it is $J_1(f(0)+2f(1)+3f(2)+...)$.

Now, if I increase $f(3)$ by 1, and decrease $f(4)$ by 1, then the sum $B$ remains the same, but $A$ changes by $+3-4=-1$. So knowing $B$ (and C) is not enough to know $A$.