Difference between revisions of "004 Sample Final A, Problem 7"
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! Foundations | ! Foundations | ||
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| − | | | + | |1) What is the formula for the nth term of an arithmetic sequence? |
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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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|Answer: | |Answer: | ||
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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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| + | |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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! Step 1: | ! Step 1: | ||
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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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! Step 2: | ! Step 2: | ||
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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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! Step 3: | ! Step 3: | ||
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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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! Final Answer: | ! Final Answer: | ||
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| − | | | + | |<math>-370</math> |
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[[004 Sample Final A|<u>'''Return to Sample Exam</u>''']] | [[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} |