<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] Do you know the rule for the sum of a finite geometric series?
$$1 + a + a^2 + \cdots + a^n = \frac{a^{n+1}-1}{a-1}$$
Now take $a=7$:
$$\begin{align}
1 + 7 + 7^2 + \cdots + 7^n\hphantom{+7^{n+1}} &= \color{maroon}{\frac{7^{n+1}-1}{6}} \\
1 + 7 + 7^2 + \cdots + 7^n+7^{n+1} &= \color{darkblue}{\frac{7^{n+2}-1}{6}} \\
\end{align}
$$
The second line is the same as the first line, but with $7^{n+1}$ added:
$$\color{maroon}{\frac{7^{n+1}-1}{6}}  + 7^{n+1} = \color{darkblue}{\frac{7^{n+2}-1}{6}}$$
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