<s>[INST] Finding explicit formula Find an explicit formula for $a_k$ if $a_0=0$ and $a_{k+1}=a_k+2^k$ for $k\geq 0$.
Thanks much for the help!
 [/INST] So
$$\begin{align}
a_k =& a_{k-1} + 2^{k-1} \\ &=  a_{k-2} + 2^{k-2}+2^{k-1} \\ &= a_{k-3} +2^{k-3} + 2^{k-2} + 2^{k-1} \\ &= \dots \\
&= a_0 + 2^0 + 2^1 + 2^2 + \dots + 2^{k-2} + 2^{k-1} \\
&= 1 + 2^1 + 2^2 + \dots + 2^{k-2} + 2^{k-1}
\end{align}
$$
What do you end up with?
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