Difference between revisions of "MR a1 Loesung Fossy"
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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 <math>q_j(i)</math> - dieses Polynom (ich brauche 7 verschidene solche) hat an der Stelle <math>j</math> den Wert 1 - an den anderen (ganzzahligen) Stellen hat es den Wert 0: | 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 <math>q_j(i)</math> - dieses Polynom (ich brauche 7 verschidene solche) hat an der Stelle <math>j</math> den Wert 1 - an den anderen (ganzzahligen) Stellen hat es den Wert 0: | ||
| − | <math>q_j(x) = { \prod_{k \in \{0 .. 6\} - j}{(x-k)} \over \prod_{\{0 .. 6\} - j}{(j-k)}}</math> | + | <math>q_j(x) = { \prod_{k \in \{0 .. 6\} - j}{(x-k)} \over \prod_{k \in \{0 .. 6\} - j}{(j-k)}}</math><br/> |
| + | |||
| + | Das gesuchte <math>p(x)</math> ist dann blos die Summe der geiegneten q's mal dem gewünschten Wert an der jeweiligen Stelle: | ||
| + | |||
| + | <math>p(x) = 1\cdot q_0(x) + 11 \cdot q_1(x) + 21 \cdot q_2(x) + ... + 13112221 \cdot q_6(x)</math><br/> | ||
Revision as of 10:13, 25 January 2009
...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. 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 verschidene 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)}