Find a particular and then the general solution for the recurrence relation $a_n = 7\cdot a_{nā1} ā 30 \cdot 2^n$
Trying to understand this equation.... We have been given a general formula for this format of the equation $2^n\cdot p = 7 \cdot 2^{n-1} - 30 \cdot 2^n$ but don't know how to plug these in. Could someone help please? Many thanks
I'd write this as $$ a_{n+1}=7a_n-30\cdot2^{n+1} $$ and compute $$ a_{n+2}=7a_{n+1}-30\cdot2^{n+2} $$ But $$ 30\cdot2^{n+2}=2\cdot(30\cdot2^{n+1})=2(7a_n-a_{n+1}) $$ so we get $$ a_{n+2}=7a_{n+1}-2(7a_n-a_{n+1}) $$ or $$ a_{n+2}-9a_{n+1}+14a_n=0 $$ which is a linear recurrence relation and the roots of the characteristic polynomial are $2$ and $7$.
