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a)

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$a_6 = a_3 \\cdot k^{6-3} = a_3 \\cdot k^3$

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$\\frac {a_6}{a_3} = k^3$    - - >   $\\frac {\\frac 1 4}{2} = k^3$    - - >    $k = \\frac 1 2$

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$a_3 = a_1 \\cdot k^2$    - - >   $ 2 = a_1 \\cdot \\frac 1 4$    - - >   $a_1 = 2 \\cdot 4 = 8$

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$a_5 = a_1 \\cdot k^4 = 8 \\cdot \\frac 1 {16} = \\frac 1 2$

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$a_{13} = a_1 \\cdot k^{12} = 8 \\cdot \\frac 1 {4096} = \\frac 1 {512}$

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b)

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$S - S_n = a_1 \\frac{1}{1-k} - a_1 \\frac{k^n-1}{k-1} = a_1 \\frac {k^n}{1-k}$

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$S - S_n = 8  \\frac {(\\frac 1 2)^n}{1-\\frac 1 2} = \\frac {16}{2^n}$

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Løser likningen  $S - S_n = \\frac 1 {1000}$

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$\\frac {16}{2^n} = \\frac 1 {1000}$

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$2^n = 16000$

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$n = \\log_2 16000 = 13.97$

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Minste verdi for n er 14.

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Finn leddene:

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$a_1$ = [[0]] 
$a_5$ = [[1]] 
$a_{13}$ = [[2]]

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Formelen for det n-te leddet i en geometrisk rekke med kvotient $k$ er

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$a_n = a_1 \\cdot k^{n-1}$

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Hvor mange ledd $n$ må vi ta med for at $S - S_n \\lt \\frac 1 {1000}$ ?
($S$ er summen av den uendelige rekka, $S_n$ er n-te delsum)

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Vi må ta med minst [[0]] ledd.

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Formelen for n-te delsum i en geometrisk rekke er $S_n = a_1 \\frac{k^n-1}{k-1}$

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Formelen for summen av en uendelig geometrisk rekke er $S = a_1 \\frac{1}{1-k}$

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I en uendelig geometrisk rekke er $a_3 = 2$ og $a_6 = \\frac 1 4$.

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I en uendelig geometrisk rekke er $a_3 = 2$  og $a_6 = \\frac 1 4$.

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