<s>[INST] Confusing algebra rule: why $\frac{7^{n+1}-1}{6} + 7^{n+1} = \frac{7^{n+2}-1}{6}$? Math rule I don't understand.
My discrete math midterm is tomorrow and I'm studying proof styles. I came across a rule (algebra maybe?) I don't quite understand and I was hoping someone could explain it step by step for me.
$$\frac{7^{n+1}-1}{6} + 7^{n+1} = \frac{7^{n+2}-1}{6}$$
I guess I can memorize it, but could someone show me how it works step by step?
Thanks
 [/INST] This isn't the sort of rule you need to memorize, but you do need know the operations to get from one side of the equation to the other.
$$
\begin{align}
\frac{7^{n+1} - 1}{6} + 7^{n+1}
&= \frac{7^{n+1} - 1}{6} + \frac{6\cdot 7^{n+1}}{6} \\
&= \frac{7^{n+1} - 1 + 6 \cdot 7^{n+1}}{6} \\
&= \frac{7 \cdot 7^{n+1} - 1}{6} \\
&= \frac{7^{n+2} -1}{6}.
\end{align}
$$
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