calculate this problem

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marinaa11

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Hi, I am struggling with this problem as I don't get what to do with the first parentheses.... I tried to find is it arithmetic progression or geometric progression... but it doesn't seem like that. Could you, please, give any hint? Thank you.

1611996508180.png

My work:

1611996539773.png
 
marinaa11 said:
Hi, I am struggling with this problem as I don't get what to do with the first parentheses.... I tried to find is it arithmetic progression or geometric progression... but it doesn't seem like that. Could you, please, give any hint? Thank you.

View attachment 24760

My work:

View attachment 24761
Click to expand...
Hint:

Multiply the first term by (√2 -1)/(√2 -1)

Multiply the second term by (√3-√2)/(√3 -√2)

and so on ....

Don't use calculator ... it will mess you up !!!
 
Good. In your first circle you should have -1 not 1.

Can you see what's going to happen if you add all the expressions you have circled?

You will need to work out the last term the the big set of brackets to see what you will be left with.
 
Harry_the_cat said:
Good. In your first circle you should have -1 not 1.

Can you see what's going to happen if you add all the expressions you have circled?

You will need to work out the last term the the big set of brackets to see what you will be left with.
Click to expand...

Thank you for the note :)
I got 1 if I add all the expressions I have circled, and I have calculated that big expression with roots, but what that multipoint mean? And sorry if I did not get Your idea, but I really want to figure this problem out :)

1612011715656.png
 
marinaa11 said:
Thank you for the note :)
I got 1 if I add all the expressions I have circled, and I have calculated that big expression with roots, but what that multipoint mean? And sorry if I did not get Your idea, but I really want to figure this problem out :)

View attachment 24768
Click to expand...
"..." is used to indicate that the pattern continues but all the terms in the middle were omitted to save space.
 
Beer soaked ramblings follow.
marinaa11 said:
Thank you for the note :)
I got 1 if I add all the expressions I have circled, and I have calculated that big expression with roots, but what that multipoint mean?
Click to expand...
A "picture" is sometimes more revealing.
[MATH]\sum\limits_{n = 1}^3 \frac{1}{\sqrt{n}+\sqrt{n+1}}=\frac{1}{\sqrt{1}+\sqrt{1+1}}+...+\frac{1}{\sqrt{3}+\sqrt{3+1}}[/MATH][MATH]\hspace{40 mm}=\frac{1}{\sqrt{1}+\sqrt{1+1}}+\frac{1}{\sqrt{2}+\sqrt{2+1}}+\frac{1}{\sqrt{3}+\sqrt{3+1}}[/MATH]

\sum\limits_{n = 1}^3 \frac{1}{\sqrt{n}+\sqrt{n+1}} - Wolfram|Alpha

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.
www.wolframalpha.com www.wolframalpha.com
Try to modify the input at Wolframalpha to suit your present need.
Or it could be easier with Desmos at
www.desmos.com

marinaa11 problem

www.desmos.com www.desmos.com
 
jonah2.0 said:
Beer soaked ramblings follow.

A "picture" is sometimes more revealing.
[MATH]\sum\limits_{n = 1}^3 \frac{1}{\sqrt{n}+\sqrt{n+1}}=\frac{1}{\sqrt{1}+\sqrt{1+1}}+...+\frac{1}{\sqrt{3}+\sqrt{3+1}}[/MATH][MATH]\hspace{40 mm}=\frac{1}{\sqrt{1}+\sqrt{1+1}}+\frac{1}{\sqrt{2}+\sqrt{2+1}}+\frac{1}{\sqrt{3}+\sqrt{3+1}}[/MATH]

\sum\limits_{n = 1}^3 \frac{1}{\sqrt{n}+\sqrt{n+1}} - Wolfram|Alpha

Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels.
www.wolframalpha.com www.wolframalpha.com
Try to modify the input at Wolframalpha to suit your present need.
Or it could be easier with Desmos at
www.desmos.com

marinaa11 problem

www.desmos.com www.desmos.com
Click to expand...

Thank you a lot!!!
 
Here is another way to do this problem.

[MATH]a_k = \dfrac{1}{\sqrt{k} + \sqrt{k+1}}= \dfrac{1}{\sqrt{k} + \sqrt{k + 1}} \cdot 1 = \dfrac{1}{\sqrt{k} + \sqrt{k + 1}} \cdot \dfrac{\sqrt{k} - \sqrt{k + 1}}{\sqrt{k} - \sqrt{k + 1}} =[/MATH]
[MATH]\dfrac{\sqrt{k} - \sqrt{k + 1}}{(\sqrt{k})^2 - (\sqrt{xk+ 1})^2} = \dfrac{\sqrt{k} - \sqrt{k + 1}}{k - (k + 1)} =[/MATH]
[MATH]\dfrac{\sqrt{k} - \sqrt{k + 1}}{- 1} = \sqrt{k + 1} - \sqrt{k}.[/MATH]
[MATH]\therefore \sum_{k=1}^n a_k = \sum_{k=1}^n (\sqrt{k + 1} - \sqrt{x}) = [/MATH]
[MATH]\left ( \sum_{k=1}^n \sqrt{k + 1} \right) - \left ( \sum_{k=1}^n \sqrt{k} \right ) =[/MATH]
[MATH]\left \{ \left ( \sum_{k=1}^{n-1} \sqrt{k + 1} \right ) + \sqrt{n+1} \right \} - \left \{ \sqrt{1} + \left ( \sum_{k=2}^n \sqrt{k} \right ) \right \} =[/MATH]
[MATH]\left \{ \left ( \sum_{k=1}^{n-1} \sqrt{k + 1} \right ) + \sqrt{n+1} \right \} - \left \{ \sqrt{1} + \left ( \sum_{j=1}^{n-1} \sqrt{j+1} \right ) \right \} =[/MATH]
[MATH]\left \{ \left ( \sum_{k=1}^{n-1} \sqrt{k + 1} \right ) + \sqrt{n+1} \right \} - \left \{ \sqrt{1} + \left ( \sum_{k=1}^{n-1} \sqrt{k+1} \right ) \right \} =[/MATH]
[MATH]\sqrt{n+1} - \sqrt{1} + \left \{ \left ( \sum_{k=1}^{n-1} \sqrt{k + 1} \right ) - \left ( \sum_{k=1}^{n-1} \sqrt{k + 1} \right ) \right \} = \sqrt{n + 1} - 1.[/MATH]
 
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