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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.

Contoh soal · fib(25)

Number-Theory · Sequences

Fib(25)

25

Langkah demi langkah

  1. F_0 = 0,; F_1 = 1

    Each Fibonacci number is the sum of the two before it.

  2. F_{2} = F_{1} + F_{0} = 1

  3. F_{3} = F_{2} + F_{1} = 2

  4. F_{4} = F_{3} + F_{2} = 3

  5. F_{5} = F_{4} + F_{3} = 5

  6. F_{6} = F_{5} + F_{4} = 8

  7. F_{7} = F_{6} + F_{5} = 13

  8. F_{8} = F_{7} + F_{6} = 21

  9. \vdots

  10. F_{24} = F_{23} + F_{22} = 46368

  11. F_{25} = F_{24} + F_{23} = 75025

Tampilkan jawaban
F_{25} = 75025

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Bagaimana: Recurrences and generating functions

  1. Each Fibonacci number is the sum of the two before it.

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