Is this identity involving binomial coefficients correct?

[Win_odd Dhamnekar]

Is the following identity involving binomial coefficients correct?

[math]\displaystyle\sum_{k=0}^n 2^{2k} (\displaystyle\frac13)^{\lfloor{k/n}\rfloor}(\frac23)^{n-\lfloor{k/n}\rfloor} =\displaystyle\sum_{i=0}^n\displaystyle\sum_{k=ni}^{(n+1)i-1} 2^{2k}(\displaystyle\frac13)^i (\displaystyle\frac23)^{n-i}[/math]

 

[blamocur]

I might not be familiar with this notation, but to me [imath](\frac{1}{3})^{\lfloor k/n\rfloor}[/imath] means 1/3 to the power of [imath]\lfloor k/n \rfloor[/imath] -- am I right?

 

[blamocur]

Also: what do you think about the question and why?

 

[Win_odd Dhamnekar]

Also: what do you think about the question and why?

I asked the following question in chat.g.p.t.


I got the following answer from chat.g.p.t.




In the above answer, how did chat.g.p.t. make equal freeform red colored snips 1 and 2 ? Is the above answer[imath] C= \frac{81}{20}[/imath]correct ?

 

[blamocur]

While I feel nervous to challenge ChatGPT, the above looks like garbage to me. Instead, I would look at the distribution of [imath]\log_2 (M_n^2)[/imath] first.