double summations translations: $$\sum_{i=1}^{n} $$\sum_{j=1}^{n} (1)

[676]

I need help with translating double summations relating to discrete math.

$$\sum_{i=1}^{n}
$$\sum_{j=1}^{n} (1) translates to what?


the translation goes from 1+1+1...+1
then it equals n^2? how did they even get 1+1...+1 in the first place? sorry this is all new to me.

thanks for help in advance

 

[stapel]

I need help with translating double summations relating to discrete math.

$$\sum_{i=1}^{n}
$$\sum_{j=1}^{n} (1) translates to what?

I'm sorry, but I don't understand your formatting above...? Do you mean the following?

. . . . .\(\displaystyle \displaystyle \sum_{i=1}^n\, \sum_{j=1}^n\, 1\)

When you ask "translates to what?", do you mean "evaluates" or "simplified to closed-form"?

the translation goes from 1+1+1...+1
then it equals n^2? how did they even get 1+1...+1 in the first place?

What did you get when you evaluated the inner summation? You have this:

. . . . .\(\displaystyle \displaystyle \sum_{j=1}^n\, 1\)

So, for j = 1, the term a_j = a₁ = 1. Then, for j = 2, the term a₂ = 1. Then, for j = 3, the term a₃ = 1. And so forth. Then you add all the terms. What then would be the result? And since you're adding n terms (since j counts off from 1 to n), what value will you get?

Then you take this value, and plug it into the outer summation. What is the value of each term? How many of these terms are there? What then is the sum?

Please show all of your work in answering the above questions. Thank you!

 

[676]

yeah i mean how does it simplify to n^2..

 

[676]

basically i understand how to evaluate but I am so confused as to how one would simplify it. like what is the difference between determining the nth sum vs nth term? I was reading

this
http://math.stackexchange.com/questions/431865/simple-double-summation

and saw the op originally was evaluating (which is all i know how to do) but then don't understand how he was getting the nth term answers.

Sorry for my confusion and I appreciate your time.

 

[stapel]

basically i understand how to evaluate but I am so confused as to how one would simplify it. like what is the difference between determining the nth sum vs nth term?

The "nth term" is the "a_j" to which I'd made reference. The "nth sum" would be the addition of all of those terms.

In your case, what is the value of every "a_j" for the first (inner) summation? Since you've got terms from a₁ to a_n, how many of those exact same value terms do you have? So when you add them up, what do you get?

Then you're taking that value (or expression) and plugging it into the second (outer) summation. What is the value (or expression) for every single term in that summation? When you add them all up, what do you get?

If you get stuck, please reply showing your work and reasoning in answering the above questions. Thank you!