Math Problem Solver

$$5,7.5,6,4.5,7,9.5,8,6.5,...$$The next term in this recurrent sequence is $a_{9}=9$.

The nth term in this sequence is given by the formula:

Recursive Formula: $$a_n=a_{n-1} - a_{n-2} + a_{n-3} + 1, \text{where } a_{1}=5, a_{2}=7.5, \text{ and } a_{3}=6$$

RECURSIVE FORMULA

Given a sequence of m terms, the recursive formula is of the form $$a_n=x_0 + x_1 a_{n-1} + ... + x_k a_{n-k}, \text{where } 1 \leq k \leq \left \lfloor \frac{m-1}{2} \right \rfloor$$

Using all the given terms, solve the systems of equations for $x_i$ when $k=1,...,\left \lfloor \frac{m-1}{2} \right \rfloor$. If $x_i$ is not found for any $k$, a recursive formula cannot be found using this method.

When $k=1$: $$a_n=x_0 + x_1 a_{n-1}, n > 1$$ Solve for $x_0$ and $x_1$: $$\begin{aligned} a_2&=x_0 + x_1 a_1 \\ a_3&=x_0 + x_1 a_2 \\ \vdots \\ a_m&=x_0 + x_1 a_{m-1}\end{aligned}$$