Geometric series

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Tore Gaupseth

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Tore Gaupseth 8 years, 9 months ago

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Tore Gaupseth 8 years, 9 months ago

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Question statement

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

Give any introductory information the student needs.

			Name	Type	Generated Value

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Parts

Part a) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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3. Gap Gap 2. Number entry
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1. Step Information only
Part b) Gap-fill
1. Gap Number entry
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1. Step Information only

Gap-fill

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Finn leddene:

$a_1$ = Gap 0.
$a_5$ = Gap 1.
$a_{13}$ = Gap 2.

Advice

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

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

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

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

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

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

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

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

$\frac {16}{2^n} = \frac 1 {1000}$

$2^n = 16000$

$n = \log_2 16000 = 13.97$

Minste verdi for n er 14.

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This question is used in the following exam:

Sumatorias y series by Luis Hernandez in Sumatorias y series.

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Taxonomy: mathcentre

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Checkpoint description

Tore Gaupseth 8 years, 9 months ago

Tore Gaupseth 8 years, 9 months ago

Error in variable testing condition

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Gap-fill