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a)
\n$a_6 = a_3 \\cdot k^{6-3} = a_3 \\cdot k^3$
\n$\\frac {a_6}{a_3} = k^3$ - - > $\\frac {\\frac 1 4}{2} = k^3$ - - > $k = \\frac 1 2$
\n$a_3 = a_1 \\cdot k^2$ - - > $ 2 = a_1 \\cdot \\frac 1 4$ - - > $a_1 = 2 \\cdot 4 = 8$
\n$a_5 = a_1 \\cdot k^4 = 8 \\cdot \\frac 1 {16} = \\frac 1 2$
\n$a_{13} = a_1 \\cdot k^{12} = 8 \\cdot \\frac 1 {4096} = \\frac 1 {512}$
\nb)
\n$S - S_n = a_1 \\frac{1}{1-k} - a_1 \\frac{k^n-1}{k-1} = a_1 \\frac {k^n}{1-k}$
\n$S - S_n = 8 \\frac {(\\frac 1 2)^n}{1-\\frac 1 2} = \\frac {16}{2^n}$
\nLøser likningen $S - S_n = \\frac 1 {1000}$
\n$\\frac {16}{2^n} = \\frac 1 {1000}$
\n$2^n = 16000$
\n$n = \\log_2 16000 = 13.97$
\nMinste verdi for n er 14.
", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "Finn leddene:
\n$a_1$ = [[0]]
$a_5$ = [[1]]
$a_{13}$ = [[2]]
Formelen for det n-te leddet i en geometrisk rekke med kvotient $k$ er
\n$a_n = a_1 \\cdot k^{n-1}$
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($S$ er summen av den uendelige rekka, $S_n$ er n-te delsum)
Vi må ta med minst [[0]] ledd.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "Formelen for n-te delsum i en geometrisk rekke er $S_n = a_1 \\frac{k^n-1}{k-1}$
\nFormelen for summen av en uendelig geometrisk rekke er $S = a_1 \\frac{1}{1-k}$
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "14", "minValue": "14", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "statement": "I en uendelig geometrisk rekke er $a_3 = 2$ og $a_6 = \\frac 1 4$.
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