[Math] Use the iteration to make an educated guess at an explicit formula for a sequenc

Question

the sequence $\{a_n\}$ is defined by recursively as $a_0=3$ and $a_n=6a_{n-1}+2$ for all $n\geq0$

Use the iteration to make an educated guess at an explicit formula for a sequence

My attempt: $a_0=3$

then $a_1=6a_0+2=20$

$a_1=122$

$a_2=734$

adding all the terms implies

3+20+122+734+…..+(upto n terms)
I am stuck form here find an explicit formula for a sequenc

Answer 1

Study the sequence $a_n - a_{n-1}$. It goes like $17, 102, 612$ etc. Do you see that this is a geometric series, with first term $17$ and constant ratio $6$?

Hence, $a_n- a_{n-1} = 17 \cdot 6^n$, you could say? Now figure out the general formula for $a_n$, and see if the additions work out. I'l give the answer in this "hidden hint":

$a_n = \dfrac{17\cdot 6^n - 2}{5}$. The sum, up to $n$ terms, is $\dfrac{17\cdot 6^{n+1} -10n-27}{25}$.