MR a1 Loesung Fossy
...und wie geht's weiter?
Gegeben sind 7 Werte - die ersten 7 Werte. Gesucht ist eine Regel für die weiteren Werte. Nichts liegt näher, als das über ein Polynom zu lösen. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p(i)} wobei Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} die gewünschte Zeile ist. Wir kennen Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(0)} ... Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(6)} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(0) = 1}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(1) = 11}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(2) = 21}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(3) = 1211}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(4) = 111221}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(5) = 312211}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(6) = 13112221}
Also gesucht ist ein Polynom, das genau das oben stehende erfüllt - sonst nix. Die Suche ist einfach, wenn man andere Polynome addiert. Ich nenne sie Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_j(i)} - dieses Polynom (ich brauche 7 verschiedene solche) hat an der Stelle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} den Wert 1 - an den anderen (ganzzahligen) Stellen hat es den Wert 0:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_j(x) = { \prod_{k \in \{0 .. 6\} - j}{(x-k)} \over \prod_{k \in \{0 .. 6\} - j}{(j-k)}}}
Das gesuchte Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)} ist dann blos die Summe der geiegneten q's mal dem gewünschten Wert an der jeweiligen Stelle:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) = 1\cdot q_0(x) + 11 \cdot q_1(x) + 21 \cdot q_2(x) + ... + 13112221 \cdot q_6(x)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_0(x)={{(x-1)(x-2)(x-3)(x-4)(x-5)(x-6)} \over { (0-1)(0-2)(0-3)(0-4)(0-5)(0-6)}} = {{x^6-21x^5+175x^4-735x^3+1624x^2-1764x+720}\over 720}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_1(x)={{(x-0)(x-2)(x-3)(x-4)(x-5)(x-6)} \over {(1-0)(1-2)(1-3)(1-4)(1-5)(1-6)}}={{x^6-20x^5+155x^4-580x^3+1440x^2-720x}\over -120}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_2(x)={{(x-0)(x-1)(x-3)(x-4)(x-5)(x-6)}\over {(2-0)(2-1)(2-3)(2-4)(2-5)(2-6)}}={{x^6-19x^5+137x^4-461x^3+702x^2-360x}\over 48 }}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q_3(x)={{(x-0)(x-1)(x-2)(x-4)(x-5)(x-6)}\over {(3-0)(3-1)(3-2)(3-4)(3-5)(3-6)}}={{x^6-18x^5+121x^4-372x^3+508x^2-240x}\over -36}}