Strategies for Solving Complex Addition Problems

[Helly123]

Homework Statement

Homework Equations

The Attempt at a Solution

I tried to rationalize the fractions by multiplied it by $$\sqrt{n + 1} - \sqrt{n} $$
it will be sum of $$ \frac {\sqrt{n + 1} - \sqrt{n}} {2n +1} $$

also tried to see the arithmetic structure
$$ 1 + \frac{1}{\sqrt{2} + \sqrt{1}} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{4} + \sqrt{3}} ... \frac{1}{\sqrt{121} + \sqrt{120}} $$

is there any method to solve it? that I don't know

 

[Dick]

Homework Statement

View attachment 218012

Homework Equations

The Attempt at a Solution

I tried to rationalize the fractions by multiplied it by $$\sqrt{n + 1} - \sqrt{n} $$
it will be sum of $$ \frac {\sqrt{n + 1} - \sqrt{n}} {2n +1} $$

also tried to see the arithmetic structure
$$ 1 + \frac{1}{\sqrt{2} + \sqrt{1}} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{4} + \sqrt{3}} ... \frac{1}{\sqrt{121} + \sqrt{120}} $$

is there any method to solve it? that I don't know

$2n+1$ is not equal to the product of $\sqrt{n + 1} - \sqrt{n}$ and $\sqrt{n + 1} + \sqrt{n}$. Fix that up and then think about it again.

 

[Helly123]

$2n+1$ is not equal to the product of $\sqrt{n + 1} - \sqrt{n}$ and $\sqrt{n + 1} + \sqrt{n}$. Fix that up and then think about it again.

thank you...

 

[SammyS]

thank you...

So ...
What did you get for a result ?

 

[Helly123]

So ...
What did you get for a result ?

11 ：）