Editing Recurrence relation (section)

===Solving first-order non-homogeneous recurrence relations with variable coefficients===
Moreover, for the general first-order non-homogeneous linear recurrence relation with variable coefficients:

:<math>a_{n+1} = f_n a_n + g_n, \qquad f_n \neq 0,</math>
there is also a nice method to solve it:<ref>{{cite web |url=http://faculty.pccu.edu.tw/%7Emeng/Math15.pdf |title=Archived copy |access-date=2010-10-19 |url-status=live |archive-url=https://web.archive.org/web/20100705023731/http://faculty.pccu.edu.tw/~meng/Math15.pdf |archive-date=2010-07-05 }}</ref>
:<math>a_{n+1} - f_n a_n = g_n</math>
:<math>\frac{a_{n+1}}{\prod_{k=0}^n f_k} - \frac{f_n a_n}{\prod_{k=0}^n f_k} = \frac{g_n}{\prod_{k=0}^n f_k}</math>

:<math>\frac{a_{n+1}}{\prod_{k=0}^n f_k} - \frac{a_n}{\prod_{k=0}^{n-1} f_k} = \frac{g_n}{\prod_{k=0}^n f_k}</math>

Let 
:<math>A_n = \frac{a_n}{\prod_{k=0}^{n-1} f_k},</math>
Then 
:<math>A_{n+1} - A_n = \frac{g_n}{\prod_{k=0}^n f_k}</math>
:<math>\sum_{m=0}^{n-1}(A_{m+1} - A_m) = A_n - A_0 = \sum_{m=0}^{n-1}\frac{g_m}{\prod_{k=0}^m f_k}</math>
:<math>\frac{a_n}{\prod_{k=0}^{n-1} f_k} = A_0 + \sum_{m=0}^{n-1}\frac{g_m}{\prod_{k=0}^m f_k}</math>
:<math>a_n = \left(\prod_{k=0}^{n-1} f_k \right) \left(A_0 + \sum_{m=0}^{n-1}\frac{g_m}{\prod_{k=0}^m f_k}\right)</math>

If we apply the formula to <math>a_{n+1} = (1 + h f_{nh}) a_n + hg_{nh}</math> and take the limit <math>h \to 0</math>, we get the formula for first order [[linear differential equation]]s with variable coefficients; the sum becomes an integral, and the product becomes the exponential function of an integral.