Is this identity involving binomial coefficients correct?

[Win_odd Dhamnekar]

Is the following identity involving binomial coefficients correct?

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

 

[blamocur]

I might not be familiar with this notation, but to me $(\frac{1}{3})^{\lfloor k/n\rfloor}$ means 1/3 to the power of $\lfloor k/n \rfloor$ -- 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$C= \frac{81}{20}$correct ?

 

[blamocur]

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