<s>[INST] Find the value of $ [1/ 3] + [2/ 3] + [4/3] + [8/3] +\cdots+ [2^{100} / 3]$ Assume that [x] is the floor function. I am not able to find any patterns in the numbers obtained. Any suggestions?
$$[1/ 3] + [2/ 3] + [4/3] + [8/3] +\cdots+ [2^{100} / 3]$$
 [/INST] Starting from $n=0$, even-$n =2k$ terms are $(4^k - 1)/3$ and the subsequent odd-$n = 2k+1$ terms are $2\times (4^k- 1)/3$. So overall:
\begin{align}
\sum_{k=0}^{49}\left[(1+2)\frac{4^{k} - 1}{3}\right] + \frac{4^{50}-1}{3} &= \sum_{k=0}^{49}(4^{k} - 1) + \frac{4^{50}-1}{3} \\
& = \sum_{k=0}^{49}(4^{k}) + \frac{4^{50}-1}{3}-50 \\
&=  \frac{4^{50}-1}{3}+ \frac{4^{50}-1}{3}-50 \\
&= \frac{2}{3}(4^{50}-1)-50 \\
&=845100400152152934331135470200 \tag{W|A}
\end{align}
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