So this question is from the manga series, “Assassination Classroom”. The Japanese Roughly above given on the cyborg translates to: “Find the general solution for a[n] as defined in following recurrence-relation sequence.”
The recurrence relation sequence in this scan looks something is shown at the bottom of that cyborg
Here’s a drawing of mine trying to depict said recurrence relation clearly.
I’m not sure if something like that would count as a recurrence relation or if the vertical “=“ signs are actually quotations (“ “)
Anyway, a character “solves” the question and has written the solution (or at least, a trick) on a grenade.: a[n+1]/3[n+1] and a[n]/3[n]
So, can somebody help me interpret what the recurrence relation in the question is and what the steps are to finding the solution?
Thank you.
Sometimes japanese is written vertically top to bottom, right to left. So the recurrence in the manga is $$ \begin{align} a_{1} &= 5, \\ a_{n+1} &= 2a_{n} + 3. \end{align} $$ I don't see any relation between the symbols on the grenade and the solution the recurrence.
This can be solved by subtituting $a_{n}-2a_{n-1}$ into $3,$ which yields $$ a_{n+1} = 3a_n - 2a_{n-1}. $$ This is a homogeneous linear recurrence, here is one way to solve it. The final solution of this recurrence is $$ a_{n} = 2^{n+2}-3. $$