Difference between revisions of "004 Sample Final A, Problem 7"

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<span class="exam"> Given a sequence <math>10, 7, 4, 1, \ldots</math> use formulae on the back page to compute <math>S_{20}</math>
 
<span class="exam"> Given a sequence <math>10, 7, 4, 1, \ldots</math> use formulae on the back page to compute <math>S_{20}</math>
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Foundations
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|-
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|1) What is the formula for the nth term of an arithmetic sequence?
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|-
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|2) What is the formula for the sum of the first n terms of an arithmetic sequence?
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|-
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|Answer:
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|-
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|1) The nth term of an arithmetic sequence is <math>A_n=A_1+d(n-1)</math> where <math>A_1</math> is the first term of the sequence and <math>d</math> is the common difference.
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|-
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|2) The sum of the first n terms of an arithmetic sequence is <math> S_n=\frac{n}{2}(A_1+A_n)</math>.
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|}
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Solution:
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 1:
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|-
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|The first term in the arithmetic sequence is <math> A_1=10</math> and the common difference is <math>d=-3</math>.
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|}
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{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 2:
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|-
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|The 20th term of this sequence is <math>A_{20}=10+-3(20-1)=10-3(19)=10-57=-47</math>.
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|}
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{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 3:
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|-
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|The sum of the first twenty terms is <math>S_{20}=\frac{20}{2}(10+-47)=10(-37)=-370</math>.
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|}
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{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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! Final Answer:
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|-
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|<math>-370</math>
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|}
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[[004 Sample Final A|<u>'''Return to Sample Exam</u>''']]

Latest revision as of 20:26, 3 May 2015

Given a 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 10, 7, 4, 1, \ldots} use formulae on the back page to compute 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_{20}}

Foundations
1) What is the formula for the nth term of an arithmetic sequence?
2) What is the formula for the sum of the first n terms of an arithmetic sequence?
Answer:
1) The nth term of an arithmetic sequence 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=A_1+d(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_1} is the first term of the sequence 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 d} is the common difference.
2) The sum of the first n terms of an arithmetic sequence 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 S_n=\frac{n}{2}(A_1+A_n)} .


Solution:

Step 1:
The first term in the arithmetic sequence 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_1=10} and the common difference 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 d=-3} .
Step 2:
The 20th term of this sequence 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_{20}=10+-3(20-1)=10-3(19)=10-57=-47} .
Step 3:
The sum of the first twenty terms 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 S_{20}=\frac{20}{2}(10+-47)=10(-37)=-370} .
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 -370}

Return to Sample Exam

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