Om Swastyastu.
\section{Rumus Barisan Kuadratik}
Apabila $u_n, a, b, c \in \mathbb{C}$ dan $n \in \mathbb{N}$, serta
\[ u_n := an^2 + bn + c, \]
maka didefinisikan
\[ v_n := u_{n + 1} - u_n \]
serta
\[ w_n := v_{n + 1} - v_n. \]
Oleh karena itu,
\[ u_1 = a + b + c, \]
\[ u_2 = 4a + 2b + c, \]
\[ u_3 = 9a + 3b + c, \]
\[ v_1 = 3a + b, \]
\[ v_2 = 5a + b, \]
\[ w_1 = 2a \]
sehingga
\[ a = (1/2)w_1, \]
\[ v_1 = (3/2)w_1 + b, \]
\[ b = v_1 - (3/2)w_1, \]
\[ u_1 = (1/2)w_1 + [v_1 - (3/2)w_1] + c, \]
\[ c = u_1 - v_1 + w_1, \]
\[ u_n = (1/2)w_1n^2 + [v_1 - (3/2)w_1]n + [u_1 - v_1 + w_1]. \]
Oleh karena itu,
\[ u_n = (1/2)w_1(n^2 - 3n + 2) + v_1(n - 1) + u_1 \]
alias
\[ u_n = (1/2)w_1(n - 1)(n - 2) + v_1(n - 1) + u_1. \]
Dengan demikian, diperoleh rumus barisan kuadratik.
Om santi santi om.
