Hi, can someone please explain why
simplifies to
and not
as I had figured out?
I'm fully confused on it
I think I know where you made the mistake...
Ok, trick question: What is $\displaystyle 3(3^{n-1})$? If you answered $\displaystyle 9^{n-1}$, you made a mistake.
Remember the rules on multiplying things with exponents with the same base?
$\displaystyle x^2*x$ is the same as $\displaystyle x^2*x^1$, of course. And since you're multiplying, you need to add the exponents. So $\displaystyle x^2*x = x^3$.
So, in fact, $\displaystyle 3(3^{n-1}) = 3^{n-1+1} = 3^n$. Which makes your answer $\displaystyle 3^n - 2$ and definitely not $\displaystyle 9^{n-1} - 2$.
Hope that clears it up! Easy mistake to make.