<s>[INST] Proof for formula for sum of sequence $3 + 9 + 27 + 81...+ 3^n$ Trying to formulate a proof for that sequence as practice. After reading this question's answer and lecture on this, I decided to try and practice with this sequence.
My try:
Base case $n = 1$
$S(n) = 3^n $
Induction step
$S(n+1) = 3^{n+1}$
Then
$Sn + n+1 = S(n+1)$
 [/INST] $$\begin{align}
\require{cancel}S & =3+9+27+\dots3^n \\
3S & =3(3+9+27+\dots3^n) \\
& = 9+27+81+\dots3^{n+1} \\
\hline S-3 & =(3+9+27+\dots3^n)-3 \\
& = 9+27+\dots3^n \\
\hline 3S-(S-3) & = \left(\color{blue}{9+27+\dots3^n}+3^{n+1}\right)-\left(\color{blue}{9+27+\dots3^n}\right) \\
2S+3 & = \left(\xcancel{9+27+\dots3^n}+3^{n+1}\right)-\left(\xcancel{9+27+\dots3^n}\right) \\
2S+3 & = 3^{n+1} \\
2S & = 3^{n+1}-3 \\
S & = \frac12\left(3^{n+1}-3\right)
\end{align}$$


$$3+9+27+\dots3^n=\frac12\left(3^{n+1}-3\right)$$

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