Difference between revisions of "005 Sample Final A, Question 22"

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(Created page with "''' Question ''' Consider the following sequence, <br> <center><math> -3, 1, -\frac{1}{3}, \frac{1}{9}, -\frac{1}{27}, \cdots </math></center><br>      a....")
 
 
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&nbsp;&nbsp;&nbsp;&nbsp; a. Determine a formula for <math>a_n</math>, the n-th term of the sequence. <br>
 
&nbsp;&nbsp;&nbsp;&nbsp; a. Determine a formula for <math>a_n</math>, the n-th term of the sequence. <br>
 
&nbsp;&nbsp;&nbsp;&nbsp; b. Find the sum <math> \displaystyle{\sum_{k=1}^\infty a_k}</math>
 
&nbsp;&nbsp;&nbsp;&nbsp; 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;"
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! Final Answers
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!Foundations
 
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|a) False. Nothing in the definition of a geometric sequence requires the common ratio to be always positive. For example, <math>a_n = (-a)^n</math>
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|1) What type of series is this?
 
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|b) False. Linear systems only have a solution if the lines intersect. So y = x and y = x + 1 will never intersect because they are parallel.
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|2) Which formulas, about this type of series, are relevant to this question?
 
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|c) False. <math>y = x^2</math> does not have an inverse.
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|3) In the formula there are some placeholder variables. What is the value of each placeholder?
 
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|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
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|Answer:
 
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|e) True.
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|1) This series is geometric. The giveaway is there is a number raised to the nth power.
 
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|f) False.  
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|2) The desired formulas are <math>a_n = a\cdot r^{n-1}</math> &nbsp; and &nbsp; <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;"
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! Step 1:
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| The sequence is a geometric sequence. The common ratio is <math>r=\frac{-1}{3}</math>.
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! 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;"
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! 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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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Final Answer:
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| <math>a_n=(-3)\left(\frac{-1}{3}\right)^{n-1}</math>
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|<math>\frac{-9}{4}</math>
 
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Latest revision as of 20:23, 21 May 2015

Question Consider the following sequence,

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 }


     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}}
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