How do I solve the summation of a sequence for my homework?

[S.R]

Homework Statement

What is the sum of:

Homework Equations

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.

 

[tiny-tim]

Hi S.R.!

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

but what is 1/(√n + √n+1) ? ​

 

[S.R]

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).

 

[Infinitum]

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]

another hint:

nobody likes square-roots on the bottom

nobody minds square-roots on the top ​

 

[S.R]

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.

 

[Infinitum]

Oh of course, √(n+1)-√n.

Yep! Now, what did you do next?

Edit : maybe its just too simple from here

 

[S.R]

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?

 

[Infinitum]

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]

Simplifying the general expression, T(n)=(sqrt(n+1)-sqrt(n))^2.

nooo, T_n = (√(n+1) - √n)/√n√(n+1) = … ?

 

[S.R]

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.

 

[Infinitum]

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]

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!

 

[S.R]

no!

Im unsure why not? I am not sure if my division is correct though.

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

 

[Infinitum]

Im unsure why not? I am not sure if my division is correct though.

Its wrong

Tn = \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n}\sqrt{n+1}}Give it another go, and write out the answer

 

[S.R]

Its wrong

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)?

 

[Infinitum]

Tn=1/sqrt(n)-1/sqrt(n+1)?

Yep. Now apply the logic I suggested in post #9 and #12.

 

[S.R]

Yep. Now apply the logic I suggested in post #9 and #12.

I don't notice any patterns to find the sum?

 

[micromass]

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??

 

[S.R]

Can you write out the first ten terms of the sum to see if you notice anything??

The sum is 9/10. Correct?

 

[S.R]

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

 

[tiny-tim]

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) ?

 

[S.R]

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]

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.

√n√(n+1)(√n + √(n+1)) = n√(n+1) + (n+1)√n

 

[S.R]

√n√(n+1)(√n + √(n+1)) = n√(n+1) + (n+1)√n

Thanks!