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

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| The sequence is a geometric sequence. The common ratio is <math>r=\frac{-1}{3}</math>.
 
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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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| 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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| <math>a_n=(-3)\left(\frac{-1}{3}\right)^{n-1}</math>
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Revision as of 13:03, 20 May 2015

Question Consider the following sequence,


     a. Determine a formula for , 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}}

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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