Tính 1/(1+√5 ) + 1/(√9 + √5) + 1/(√9+√13) + ... + 1/(√2013 + √2017)

To simplify the expression, we can rationalize the denominators of each fraction.

For the first fraction, we have:

1/(1+√5) = (1-√5)/(1+√5)(1-√5) = (1-√5)/(1-5) = (1-√5)/(-4) = (√5-1)/4

For the second fraction, we have:

1/(√9 + √5) = (√9 - √5)/(√9 + √5)(√9 - √5) = (√9 - √5)/(9 - 5) = (√9 - √5)/4 = (√9 - √5)/2

For the third fraction, we have:

1/(√9 + √13) = (√9 - √13)/(√9 + √13)(√9 - √13) = (√9 - √13)/(9 - 13) = (√9 - √13)/(-4) = (√13 - √9)/4

Continuing this pattern, we can see that each fraction can be simplified to (√n+2 - √n)/4.

Now, let's look at the sum of all these fractions:

(√5-1)/4 + (√9-√5)/4 + (√13-√9)/4 + ... + (√2017-√2013)/4

Notice that the terms (√n+2 - √n) and (√n+6 - √n+2) cancel each other out:

(√5-1)/4 + (√9-√5)/4 + (√13-√9)/4 + ... + (√2017-√2013)/4
= (√5-1)/4 + (√9-√5)/4 + (√13-√9)/4 + ... + (√2017-√2013)/4 + (√2021-√2017)/4 - (√2017-√2013)/4

We can see that all the terms in the middle cancel out, leaving us with:

(√2021-1)/4

Therefore, the simplified expression is (√2021-1)/4.