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

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.
$$
\begin{align}
\frac{7^{n+1} - 1}{6} + 7^{n+1}
&= \frac{7^{n+1} - 1}{6} + \frac{6\cdot 7^{n+1}}{6} \\
&= \frac{7^{n+1} - 1 + 6 \cdot 7^{n+1}}{6} \\
&= \frac{7 \cdot 7^{n+1} - 1}{6} \\
&= \frac{7^{n+2} -1}{6}.
\end{align}
$$
