# Summation of a Sequence

## Homework Statement

What is the sum of:

N/A

## The Attempt at a Solution

I'm unsure how to start.

Note: I'm in Grade 10, so I may not have the mathematical skills necessary to understand the solutions you provide.

Any help/guidance would be appreciated.

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tiny-tim
Homework Helper
Hi S.R.!

Hint: each term is 1/√n√n+1(√n + √n+1) …

but what is 1/(√n + √n+1) ?

Hi S.R.!

Hint: each term is 1/√n√n+1(√n + √n+1) …

but what is 1/(√n + √n+1) ?
Assuming n+1 isn't inclusive: 1/(2√n+1). I'm not sure what to do with this information (if I'm correct, that is).

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Assuming n+1 isn't inclusive: 1/(2√n+1). I'm not sure what to do with this information (if I'm correct, that is).
Nope, n+1 is inclusive.

The general term is

$T_n = \frac{1}{(\sqrt{n}\sqrt{n+1})(\sqrt{n} + \sqrt{n+1})}$

But, how can you simplify this part of the above equation?

$\frac{1}{(\sqrt{n} + \sqrt{n+1})}$

Hint:Rationalize...

tiny-tim
Homework Helper
another hint:

nobody likes square-roots on the bottom

nobody minds square-roots on the top

Nope, n+1 is inclusive.

The general term is

$T_n = \frac{1}{(\sqrt{n}\sqrt{n+1})(\sqrt{n} + \sqrt{n+1})}$

But, how can you simplify this part of the above equation?

$\frac{1}{(\sqrt{n} + \sqrt{n+1})}$

Hint:Rationalize...
Oh of course, √(n+1)-√n.

Oh of course, √(n+1)-√n.
Yep! Now, what did you do next?

Edit : maybe its just too simple from here

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Yep! Now, what did you do next?

Edit : maybe its just too simple from here
Simplifying the general expression, T(n)=(sqrt(n+1)-sqrt(n))^2. However, I'm stuck here. Is there an applicable formula?

Simplifying the general expression, T(n)=(sqrt(n+1)-sqrt(n))^2. However, I'm stuck here. Is there an applicable formula?
Uhh, where'd get that whole square from??

The simplification from the general term will yield you a difference of two terms. You can write them as

$T_1 = A_2 - A_1$
$T_2 = A_3 - A_2$

and so on. The sum of all terms is the sum of the series. Can you notice something in the above equations that makes solving this easier?

tiny-tim
Homework Helper
Simplifying the general expression, T(n)=(sqrt(n+1)-sqrt(n))^2.
nooo, Tn = (√(n+1) - √n)/√n√(n+1) = … ?

Sorry I misread from my iPhone. However, the simplification from the general term is sqrt(n+1)-sqrt(n)/sqrt(n)sqrt(n+1) = sqrt(n)-sqrt(n+1).

From your explanation I noticed, Tn=An+1-An.

Note that the extra terms are suppose to be subscrippts.

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Sorry I misread from my iPhone. However, the simplification from the general term is sqrt(n+1)-sqrt(n)/sqrt(n)sqrt(n+1) = sqrt(n)-sqrt(n+1).

From your explanation I noticed, Tn=Tn+1-Tn.

Note that the extra terms are suppose to be subscrippts.
That would make $2T_n = T_{n+1}$ which is untrue. Its better to use different term letters for it. So, what do you get from that relation, by summing it all up?

tiny-tim
Homework Helper
Sorry I misread from my iPhone. However, the simplification from the general term is sqrt(n+1)-sqrt(n)/sqrt(n)sqrt(n+1) = sqrt(n)-sqrt(n+1).
no!!!!!

no!!!!!
Im unsure why not? Im not sure if my division is correct though.

Edit: Sorry, I was replying in English class and got distracted .

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Im unsure why not? Im not sure if my division is correct though.
Its wrong :tongue2:

$$Tn = \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n}\sqrt{n+1}}$$

Give it another go, and write out the answer

Its wrong :tongue2:

$$Tn = \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n}\sqrt{n+1}}$$

Give it another go, and write out the answer
Tn=1/sqrt(n)-1/sqrt(n+1)?

Tn=1/sqrt(n)-1/sqrt(n+1)?
Yep. Now apply the logic I suggested in post #9 and #12.

Yep. Now apply the logic I suggested in post #9 and #12.
I don't notice any patterns to find the sum?

I don't notice any patterns to find the sum?
Can you write out the first ten terms of the sum to see if you notice anything??

Can you write out the first ten terms of the sum to see if you notice anything??
The sum is 9/10. Correct?

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I still don't understand how you came up with the general form of the sequence, though?

tiny-tim
Homework Helper
I still don't understand how you came up with the general form of the sequence, though?
what is $$\Sigma_5^{21}\ \left(\frac{1}{n}-\frac{1}{n+1}\right)$$ ?

what is $$\Sigma_5^{21}\ \left(\frac{1}{n}-\frac{1}{n+1}\right)$$ ?
The terms 1/5 and -1/24 are left after summation, therefore 19/120. However, my question is how did you obtain: Tn=1/(sqrt(n)sqrt(n+1))((sqrt(n+1)+sqrt(n))? Sorry for the notation.

tiny-tim