Show that the total number of states available in a shell withprincipal quantum

Question

Show that the total number of states available in a shell withprincipal quantum number n is 2n2  
I started out using a summation of 2(2l+1) from l=0 to n-1. This got me 2+6+10+....
That's pretty much as far as I've gotten. I don't knowhow to get 2n2
I'm sure this is just a simple arithmetic problem. Thanks in advance for any help.

Show that the total number of states available in a shell withprincipal quantum number n is 2n2  
I started out using a summation of 2(2l+1) from l=0 to n-1. This got me 2+6+10+....
That's pretty much as far as I've gotten. I don't knowhow to get 2n2
I'm sure this is just a simple arithmetic problem. Thanks in advance for any help.


I started out using a summation of 2(2l+1) from l=0 to n-1. This got me 2+6+10+....
That's pretty much as far as I've gotten. I don't knowhow to get 2n2
I'm sure this is just a simple arithmetic problem. Thanks in advance for any help.

Explanation / Answer

let A = 2 + 6 + 10 + ... + 2(2n-1) A = 2(1 + 3 + 5 + ... 2n-1)     (there are nterms) (1) write A = 2(2n-1 + ... + 5 + 3 + 1)     (2) (1) + (2) 2A = 2(2n + 2n + 2n + ... +2n)             (there are n terms) A = 2n + 2n + 2n + ... + 2n A = 2n*n = 2n2

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