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Recurrences and generating functions
Solving recurrence relations, and encoding a sequence as a power series.
A recurrence defines aₙ from earlier terms; a generating function Σ aₙ xⁿ packs the whole sequence into one function whose algebra does the counting. Picture it: the Fibonacci numbers growing by the golden ratio each step. Think it: linear recurrences with constant coefficients are solved exactly like constant-coefficient differential equations, a characteristic polynomial.
Çözümlü örnek · fib(25)
Adım adım
- F_0 = 0,; F_1 = 1
Each Fibonacci number is the sum of the two before it.
- F_{2} = F_{1} + F_{0} = 1
- F_{3} = F_{2} + F_{1} = 2
- F_{4} = F_{3} + F_{2} = 3
- F_{5} = F_{4} + F_{3} = 5
- F_{6} = F_{5} + F_{4} = 8
- F_{7} = F_{6} + F_{5} = 13
- F_{8} = F_{7} + F_{6} = 21
- \vdots
- F_{24} = F_{23} + F_{22} = 46368
- F_{25} = F_{24} + F_{23} = 75025
Cevabı göster
Şimdi sen Bir problem seçin, ya da kendi probleminizi yazın ya da çizin. Her adım, bir resim, cevap sorulana kadar gizli.
Burada kullanılan sembolleri
Tüm tanım, resim ve her harfin anlamı için herhangi bir sembolü tıklayın.
Nasıl: Recurrences and generating functions
- Each Fibonacci number is the sum of the two before it.
Sık sorulan sorular
Permutation or combination?
Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).
What is a graph in this sense?
Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.
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