General term of recurrence relation

Question

Find the general term of the following recurrence relation:

$$a_{1} = 2$$

$$a_{n+1} = \frac{2a_{n} - 1}{3}$$

I've tried to find the first few terms:

$$a_{1} = 2$$ $$a_{2} = \frac{2 \cdot 2 - 1}{3} = 1$$ $$a_{3} = \frac{2 \cdot 1 - 1}{3} = \frac{1}{3}$$ $$a_{4} = -\frac{1}{9}$$ $$a_{5} = -\frac{1}{27}$$

but can't see any pattern, especially considering the first 2 terms.

Answer 1

We have $$ 3a_{n+1}=2a_n-1 \tag{1}$$ and we may try to get rid of the inhomogeneous term $-1$ by a suitable substitution.
For instance, by setting $a_n=b_n-1$, we get $b_1=3$ and $$ 3b_{n+1}=2b_n \tag{2} $$ from which $b_n = 3\cdot\frac{2^{n-1}}{3^{n-1}}$ readily follows. The general term of the original sequence is so: $$ a_n = \color{red}{3\cdot\frac{2^{n-1}}{3^{n-1}}-1}.\tag{3}$$

Answer 2

Following your (@Jack D'Aurizio) suggestion, here is what I understood:

The given recurrence relation: $3a_{n} = 2a_{n-1} - 1$, with initial conditions: $a_{1} = 2$ is a $1^{st}$ degree linear non-homogeneous recurrence relation.

Solution:

We are searching for solution¹ of the form: $a_{n} = b_{n} + h_{n}$ (1), where $h_{n}$ is the sequence satisfying the associated homogeneous recurrence relation and $b_{n}$ is a solution which is similar to $f(n)$.

1.The associated homogeneous recurrence relation is: $3h_{n} = 2h_{n-1}$ and its characteristic equation is: $3r - 2 = 0$ or $ r = \frac{2}{3}$ and the solution is:

$$h_{n} = \alpha_{0} (\frac{2}{3})^{n-1} $$

From the initial conditions: $a_{1} = \alpha_{0} (\frac{2}{3})^{1} = 2$ or $\alpha_{0} = 3$, so the solution of the homogeneous recurrence is:

$$h_{n} = 3 (\frac{2}{3})^{n-1} $$

2. In our case $b_{n} = -1$ and from (1) we have:

$$a_{n} = 3 (\frac{2}{3})^{n-1} - 1$$

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^{1. $a_{n}$ satisfies both the recurrence relation and the initial conditions.}

^{2. $f(n)$ - function depending only on $n$.}