$$\frac{4}{5},\frac{3}{4},\frac{16}{19},\frac{25}{26},\frac{12}{11},...$$The next term in this recurrent sequence is $a_{6}=\frac{31962847}{26031577}$.
The nth term in this sequence is given by the formula:
Explicit Formula: $$a_n=\frac{\left(-1265116428719 + 825345664 \sqrt{3416686}\right) \left(\frac{1336}{2211} - \frac{7 \sqrt{3416686}}{28743}\right)^{n}}{-616611950640 + 447773760 \sqrt{3416686}} + \left(\frac{1533265747 \sqrt{3416686}}{8119110365100} + \frac{7433905687}{4695591600}\right) \left(\frac{7 \sqrt{3416686}}{28743} + \frac{1336}{2211}\right)^{n} - \frac{35884}{25137}$$
Recursive Formula: $$a_n=\frac{2672 a_{n-1}}{2211} - \frac{4670 a_{n-2}}{28743} + \frac{35884}{546117}, \text{where } a_{1}=\frac{4}{5} \text{ and } a_{2}=\frac{3}{4}$$
RECURSIVE FORMULA
Linear Recurrence Relation
[View Steps]
Given a sequence of 5 terms, the recursive formula is of the form $$a_n=x_0 + x_1 a_{n-1} + ... + x_k a_{n-k}, \text{where } 1 \leq k \leq 2$$
When $k=2$: $$a_n=x_0 + a_{n-1} x_{1} + a_{n-2} x_{2}, n > 2$$ $$\begin{aligned}\frac{16}{19}&=x_{0} + \frac{3 x_{1}}{4} + \frac{4 x_{2}}{5}\\\frac{25}{26}&=x_{0} + \frac{16 x_{1}}{19} + \frac{3 x_{2}}{4}\\\frac{12}{11}&=x_{0} + \frac{25 x_{1}}{26} + \frac{16 x_{2}}{19} \end{aligned}$$$$\left\{ x_{0} : \frac{35884}{546117}, \ x_{1} : \frac{2672}{2211}, \ x_{2} : - \frac{4670}{28743}\right\}$$
The nth term rule is:$$a_n=\frac{2672 a_{n-1}}{2211} - \frac{4670 a_{n-2}}{28743} + \frac{35884}{546117}$$
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 $x_1$: $$\begin{aligned} a_2&=x_0 + x_1 a_1 \\ a_3&=x_0 + x_1 a_2 \\ \vdots \\ a_m&=x_0 + x_1 a_{m-1}\end{aligned}$$
EXPLICIT FORMULA
Non-Homogeneous Solution
[View Steps]
Given a non-homogeneous linear recurrence relation $$a_n=\frac{2672 a_{n-1}}{2211} - \frac{4670 a_{n-2}}{28743} + \frac{35884}{546117}$$ the closed formula where $a_h$ is the general solution of $\frac{2672 a_{n-1}}{2211} - \frac{4670 a_{n-2}}{28743}$, and $a_p$ is the particular solution of $\frac{35884}{546117}$ is $$a_n=a_h + a_p$$
Find $a_h$:
characteristic equation: $r^{2} - \frac{2672 r}{2211} + \frac{4670}{28743}=0$
roots: $\left\{ \frac{1336}{2211} - \frac{7 \sqrt{3416686}}{28743} : 1, \ \frac{7 \sqrt{3416686}}{28743} + \frac{1336}{2211} : 1\right\}, \text{ where r:m}$
general solution: $a_h=c_{1} \left(\frac{1336}{2211} - \frac{7 \sqrt{3416686}}{28743}\right)^{n} + c_{2} \left(\frac{7 \sqrt{3416686}}{28743} + \frac{1336}{2211}\right)^{n}$
Find $a_p$:
trial solution: $t(n)=A$
particular solution: $a_p=- \frac{35884}{25137}$
$\begin{aligned}A&=\frac{2672 A}{2211} - \frac{4670 A}{28743} + \frac{35884}{546117} \\ A&=- \frac{35884}{25137}\end{aligned}$
Using the first $2$ terms, solve for $c_i$: $$a_n=c_{1} \left(\frac{1336}{2211} - \frac{7 \sqrt{3416686}}{28743}\right)^{n} + c_{2} \left(\frac{7 \sqrt{3416686}}{28743} + \frac{1336}{2211}\right)^{n} - \frac{35884}{25137}$$ $$\begin{aligned}\frac{4}{5}&=c_{1} \left(\frac{1336}{2211} - \frac{7 \sqrt{3416686}}{28743}\right)^{1} + c_{2} \left(\frac{7 \sqrt{3416686}}{28743} + \frac{1336}{2211}\right)^{1} - \frac{35884}{25137}\\\frac{3}{4}&=c_{1} \left(\frac{1336}{2211} - \frac{7 \sqrt{3416686}}{28743}\right)^{2} + c_{2} \left(\frac{7 \sqrt{3416686}}{28743} + \frac{1336}{2211}\right)^{2} - \frac{35884}{25137}\end{aligned}$$ $$\left\{ c_{1} : \frac{-1265116428719 + 825345664 \sqrt{3416686}}{-616611950640 + 447773760 \sqrt{3416686}}, \ c_{2} : \frac{1533265747 \sqrt{3416686}}{8119110365100} + \frac{7433905687}{4695591600}\right\}$$
The nth term rule is: $$a_n=\frac{\left(-1265116428719 + 825345664 \sqrt{3416686}\right) \left(\frac{1336}{2211} - \frac{7 \sqrt{3416686}}{28743}\right)^{n}}{-616611950640 + 447773760 \sqrt{3416686}} + \left(\frac{1533265747 \sqrt{3416686}}{8119110365100} + \frac{7433905687}{4695591600}\right) \left(\frac{7 \sqrt{3416686}}{28743} + \frac{1336}{2211}\right)^{n} - \frac{35884}{25137}$$
Given a non-homogeneous linear recurrence relation in the form $$a_n=C_1 a_{n-1} + C_2 a_{n-2} + ... + C_k a_{n-k} + f(n), \text{where } f(n)\neq0$$the closed formula where $a_h$ is the general solution of $C_1 a_{n-1} + C_2 a_{n-2} + ... + C_k a_{n-k}$ (homogeneous) and $a_p$ is the particular solution of $f(n)$ is $$a_n=a_h + a_p$$
To find $a_h$, use the Characteristic Root Technique. [ show ]
Given a homogeneous linear recurrence relation of order $k$$$C_1 a_{n-1} + C_2 a_{n-2} + ... + C_k a_{n-k}$$ the characteristic equation is the $k$th degree polynomial equation$$r^k - C_1 r^{k-1} - C_2 r^{k-2} - ... - C_{k-1} r - C_k=0$$
Characteristic Roots
1. If $r$ is a distinct real root, then the general solution includes a term $r^n$.
2. If $r$ is a repeated real root with multiplicity of $m$, then the general solution includes the terms $r^n, n r^n,..., n^{m-1} r^n$.
3. If $r$ is a pair of complex roots, then the general solution includes the terms $|r|^n \sin(n\theta)$ and $|r|^n \cos(n\theta)$ where $r=a\pm b i= |r|e^{\pm \theta i}$.
Let the terms be denoted by $f_i(n)$ where $i=1,...,k$. The general solution is $$a_h=c_1 f_1(n) + c_2 f_2(n) + ... + c_k f_k(n)$$
To find $a_p$, find an appropriate trial solution. [ show ]
1. Choose a reasonable trial solution $t(n)$ for $f(n)$.
Possible Trial Solutions for f(n)
1. Constant
A
2. mth Degree Polynomial
Amnm + ... + A1n + A0
3. Exponential with base b
Abn OR Anbn OR etc.
2. Substitute $a_i = t(i)$ for $i = n, n-1, ..., n-k$ in the recurrence relation, and solve for the missing variables $A_j$.
3. The particular solution is $$a_p=t(n)$$
The closed formula where $c_i$ are coefficients determined by the initial conditions is $$a_n=c_1 f_1(n) + c_2 f_2(n) + ... + c_k f_k(n) + t(n)$$
Solve the systems of equations to find $c_i$: $$\begin{aligned}a_1 &= c_1 f_1(1) + c_2 f_2(1) + ... + c_k f_k(1) + t(1) \\ a_2 &= c_1 f_1(2) + c_2 f_2(2) + ... + c_k f_k(2) + t(2) \\ \vdots \\ a_k &= c_1 f_1(k) + c_2 f_2(k) + ... + c_k f_k(k) + t(k) \end{aligned}$$
