<s>[INST] How come $2 \times 3^k + 3^k = 3 \times 3^k$ I get something else
$$\begin{align*}
&2 \times 3^k + 3^k=\\
&2 \times 3^k \times 2=\\
&4 \times 3^k
\end{align*}$$
What does $3^k + 3^k$ give exactly?
 [/INST] Your calculation appears to be based on the notion that $2\times 3^k+3^k$ means $2\times(3^k+3^k)$, which would indeed be $4\cdot3^k$. However, in the absence of parentheses multiplication is performed before addition, so $2\times 3^k+3^k$ actually means $(2\times 3^k)+3^k$. This is $(2\times 3^k)+(1\times 3^k)$, which is clearly just $3\times 3^k$, or $3^{k+1}$.
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