Prove that: `1/(1+sqrt(2))+1/(sqrt(2)+sqrt(3))+1/(sqrt(3)+\ sqrt(4))+1/(sqrt(4)+sqrt(5))+1/(sqrt(5)+sqrt(6))+1/(sqrt(6)+sqrt(7))+\ 1/(sqrt(7)+sqrt(8))+1/(sqrt(8)+sqrt(9))=2`

Explanation

To prove that:

\[ \frac{1}{1+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{4}} + \frac{1}{\sqrt{4}+\sqrt{5}} + \frac{1}{\sqrt{5}+\sqrt{6}} + \frac{1}{\sqrt{6}+\sqrt{7}} + \frac{1}{\sqrt{7}+\sqrt{8}} + \frac{1}{\sqrt{8}+\sqrt{9}} = 2 \]

we will rationalize each term in the sum. Here is the step-by-step solution:

### Step 1: Rationalize each term
For each term \(\frac{1}{\sqrt{n} + \sqrt{n+1}}\), multiply the numerator and the denominator by the conjugate of the denominator \(\sqrt{n} - \sqrt{n+1}\):
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