If the average of 2 successive years’ production 1/2($a_n + a_{n-1}$) is 2n + 5 and $a_0=3$, find $a_n$.

Question

If the average of 2 successive years’ production $\frac{1}{2}(a_n + a_{n-1})$ is $2n + 5$ and $a_0=3$, find $a_n$. I started by solving for $a_n$ and got: $a_n = 4n+10-a_{n-1}$ but I am unsure how to finish the problem.

Answer 1

This is a non-homogeneous linear recurrence equation where the non-homogeneous part is a polynomial. There are general methods to solve such recurrences but your specific example can be simplified.

To solve the recurrence relation

$$ \begin{cases} a_0 = 3 \\ a_n = 4n - 10 - a_{n-1} \end{cases} $$

First notice that $a_1 = -9$. One can try to expand the recurrence one term further:

$$ a_{n+2} = 4(n+2) - 10 - a_{n+1} = 4(n+2) - 10 - (4(n+1) - 10 - a_{n}) $$ $$ a_{n+2} = 4n + 8 - 10 - 4n - 4 + a_{n} = a_{n} + 4 $$

Therefore we have $$ a_{2k} = a_0 + 4k = 3 + 2(2k) $$ $$ a_{2k+1} = a_1 + 4k = -11 + 2(2k+1)$$

Combining them, we have $a_n = 2n -4 + 7(-1)^n$.