Pick a natural number $N \geq 1$. For $k \geq 0$, where $A_{k}$ is defined as $$A_{k} = \left( |a_{2^{k}N}| + |a_{2^{k}N+1}| \right) + \left( |a_{2^{k}N+2}| + |a_{2^{k}N+3}| \right) + \cdot \cdot \cdot \left( |a_{2^{k+1}N-2}| + |a_{2^{k+1}N-1}| \right)$$
Write $\sum_{k \geq 0} A_{k}$.
I am immensely confused by this. How did $A_{k}$ went from $N+1, N+2, N+3$ to $N-2, N-1$ ? Where is the turning point?
Below is what I have tried.
Judging by the first few terms, the pattern seems to be $$\sum_{k \geq 0} A_{k} = \sum_{k \geq 0} |a_{2^{k}N+k}|$$
However, I don't understand the last two terms, if $k$ is going to $\infty$, how did $A_{k}$ end up as $$|a_{2^{k+1}N-2}| + |a_{2^{k+1}N-1}|$$
And how does this form the series $\sum_{k \geq 0} A_{k}$ ?
$A_k$ goes from $2^k N,2^k N+1,\cdots,2^{k+1}N-1$, $2^k N$ in total. Then $$\sum_{k\geq 0}A_k=\sum_{k\geq N}|a_k|$$.
Edit.
Maybe you can try $k=2,N=3$ or some others. Then $$A_0=|a_3|+|a_4|+|a_5|$$ $$A_1=|a_6|+|a_7|+\cdots+|a_{11}|$$ ...
So from the form of $A_0$ and $A_1$, you can find the sum $\sum_{k\geq 0}A_k=\sum_{k\geq 3}|a_k|=\sum_{k\geq N}|a_k|$.