Transform recursive sequence to direct. I am taking the GRE General Exam in a few weeks and there are some problems about sequences that I have found a bit difficult, e.g given a sequence in recursive form like $S_{n} = S_{n-1} - 10$  and some value for this sequence $S_{3}=0$ what is the value  $S_{25}$?
I know that the sequence in the direct form is $S_{n} = -10n + 30 $ but how this turns out?
for example can someone tell me step by step whats $S_{n} = 2S_{n-1}-4$ direct formula given that $S_{1}=6$
Thanks

First part:
$$S_n-S_{n-1}=-10$$
Summing by telescoping for $n=m$ down to $4$ gives
$$S_m-S_3=-10(m-3)\\
S_m=-10m+30 \Longleftrightarrow S_n=-10n+30\;\;\blacksquare$$
Second part:
$$\begin{align}
S_n&=2S_{n-1}-4\\
S_n-4&=2(S_{n-1}-4)\\
&=2^2 (S_{n-2}-4)\\
&=\vdots\\
&=2^{n-1}(S_1-4)\\
&=2^{n-1}(2)\\
&=2^n\\
S_n&=2^n+4\;\;\blacksquare
\end{align}$$
