Rekurrent ketma-ketliklar

🟢 — masalani tezlashtiradi · 🟡 — ilg'or daraja · 🔵 — chuqur o'rganish (olimpiada darajasi)

🟡 Iteratsiya va limit

$u_{n+1} = f(u_n)$ ketma-ketlik $L$ ga yaqinlashsa: $\;L = f(L)$

Misol: $u_{n+1} = \dfrac{1}{2}u_n + 3$ uchun $L = \dfrac{1}{2}L + 3$, ya'ni $L = 6$.

🔵 Chiziqli rekurrensiyaning yechimi

$u_{n+1} = k\,u_n + b$ ($k \ne 1$) bo'lsa:

$u_n = k^{n-1}(u_1 - L) + L, \qquad L = \frac{b}{1 - k}$