Difference between revisions of "005 Sample Final A, Question 22"
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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 -3, 1, -\frac{1}{3}, \frac{1}{9}, -\frac{1}{27}, \cdots }
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a. Determine a formula for <math>a_n</math>, the n-th term of the sequence. <br> | a. Determine a formula for <math>a_n</math>, the n-th term of the sequence. <br> | ||
b. Find the sum <math> \displaystyle{\sum_{k=1}^\infty a_k}</math> | b. Find the sum <math> \displaystyle{\sum_{k=1}^\infty a_k}</math> | ||
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| − | ! | + | !Foundations |
| + | |- | ||
| + | |1) What type of series is this? | ||
| + | |- | ||
| + | |2) Which formulas, about this type of series, are relevant to this question? | ||
| + | |- | ||
| + | |3) In the formula there are some placeholder variables. What is the value of each placeholder? | ||
| + | |- | ||
| + | |Answer: | ||
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| + | |1) This series is geometric. The giveaway is there is a number raised to the nth power. | ||
| + | |- | ||
| + | |2) The desired formulas are <math>a_n = a\cdot r^{n-1}</math> and <math>S_\infty = \frac{a_1}{1-r}</math> | ||
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| − | | | + | |3) <math>a_1</math> is the first term in the series, which is <math> -3</math>. The value for r is the ratio between consecutive terms, which is <math>\frac{-1}{3}</math> |
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| − | ! Step | + | ! Step 1: |
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| − | | | + | | The sequence is a geometric sequence. The common ratio is <math>r=\frac{-1}{3}</math>. |
|} | |} | ||
{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| − | ! Step | + | ! Step 2: |
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| − | | | + | | The formula for the nth term of a geometric series is <math>a_n=ar^{n-1}</math> where <math>a</math> is the first term of the sequence. |
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| − | | | + | | So, the formula for this geometric series is <math>a_n=(-3)\left(\frac{-1}{3}\right)^{n-1}</math>. |
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| − | ! Step | + | ! Step 3: |
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| − | | | + | | For geometric series, <math>\displaystyle{\sum_{k=1}^\infty a_k}=\frac{a}{1-r}</math> if <math>|r|<1</math>. Since <math>|r|=\frac{1}{3}</math>, |
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| − | | | + | | we have <math>\displaystyle{\sum_{k=1}^\infty a_k}=\frac{-3}{1-\frac{-1}{3}}=\frac{-9}{4}</math>. |
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! Final Answer: | ! Final Answer: | ||
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| − | | | + | | <math>a_n=(-3)\left(\frac{-1}{3}\right)^{n-1}</math> |
| + | |- | ||
| + | |<math>\frac{-9}{4}</math> | ||
|} | |} | ||
Latest revision as of 20:23, 21 May 2015
Question Consider the following sequence,
a. Determine a formula for 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 a_n}
, the n-th term of the sequence.
b. Find the sum 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 \displaystyle{\sum_{k=1}^\infty a_k}}
| Foundations |
|---|
| 1) What type of series is this? |
| 2) Which formulas, about this type of series, are relevant to this question? |
| 3) In the formula there are some placeholder variables. What is the value of each placeholder? |
| Answer: |
| 1) This series is geometric. The giveaway is there is a number raised to the nth power. |
| 2) The desired formulas are 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 a_n = a\cdot r^{n-1}} and 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 S_\infty = \frac{a_1}{1-r}} |
| 3) 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 a_1} is the first term in the series, which is 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 -3} . The value for r is the ratio between consecutive terms, which is 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 \frac{-1}{3}} |
| Step 1: |
|---|
| The sequence is a geometric sequence. The common ratio is 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 r=\frac{-1}{3}} . |
| Step 2: |
|---|
| The formula for the nth term of a geometric series is 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 a_n=ar^{n-1}} where 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 a} is the first term of the sequence. |
| So, the formula for this geometric series is 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 a_n=(-3)\left(\frac{-1}{3}\right)^{n-1}} . |
| Step 3: |
|---|
| For geometric series, 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 \displaystyle{\sum_{k=1}^\infty a_k}=\frac{a}{1-r}} if 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 |r|<1} . Since 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 |r|=\frac{1}{3}} , |
| we have 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 \displaystyle{\sum_{k=1}^\infty a_k}=\frac{-3}{1-\frac{-1}{3}}=\frac{-9}{4}} . |
| Final Answer: |
|---|
| 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 a_n=(-3)\left(\frac{-1}{3}\right)^{n-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 \frac{-9}{4}} |