Let $\displaystyle a$ be the $\displaystyle 1000$th Fibonacci number; and,

Let $\displaystyle b$ be the $\displaystyle 1001$st Fibonacci number.

Then, write the $\displaystyle 1003$rd Fib. number in terms of $\displaystyle a$ and $\displaystyle b$. Simplify.

---

So $\displaystyle F_{1000} = a, F_{1001} = b$, then

$\displaystyle a + b = F_{1002} = c$

And so $\displaystyle c+b = F_{1003}$

I doubt this is right because I'm introducing a new constant. Any ideas?