Prove that the sum $$ \sqrt{1001^2 + 1}+\sqrt{1002^2 + 1} \ + ... + \sqrt{2000^2 + 1}$$ is irrational.
The textbook has the solution too but I'm unable to understand it.
The strategy is divided into two parts:-
- Proving that the sum is not an integer
- Proving that it is a zero of a monic polynomial
First part is simple to understand and prove. $$n^2 +1 < n^2 +2 < n^2 + 2n\left(\frac{1}{n}\right) + \left(\frac{1}{n}\right)^2 = (n+\frac{1}{n})^2\,.$$ So $$S = 1001 +a_1 + 1002 + a_2 + \ ... \ +2000 + a_{1000}\,, $$ and $$0 <a_1 + a_2 + ...+a_{1000} < \frac{1}{1001} (1000) < 1\,.$$
But how is the second part going to help in proving the result?