Difference between revisions of "008A Sample Final A, Question 8"
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|1) Which of the <math>S_n</math> formulas should you use? | |1) Which of the <math>S_n</math> formulas should you use? | ||
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|The formula for <math>S_n</math> that involves a common difference, d, is the one we want. The other formula involves a common ratio, r. So we have to determine the value of n, <math>A_1</math>, and <math>A_n</math> | |The formula for <math>S_n</math> that involves a common difference, d, is the one we want. The other formula involves a common ratio, r. So we have to determine the value of n, <math>A_1</math>, and <math>A_n</math> | ||
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|Now we determine <math>A_n</math> by finding d. To do this we use the formula <math>A_n = A_1 + d(n - 1)</math> with n = 2, <math>A_1 = 27</math>, and<math>A_2 = 23</math>. This yields d = -4. | |Now we determine <math>A_n</math> by finding d. To do this we use the formula <math>A_n = A_1 + d(n - 1)</math> with n = 2, <math>A_1 = 27</math>, and<math>A_2 = 23</math>. This yields d = -4. | ||
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|Now we have d, and we can use the same formula for <math>A_n</math> to get <math>A_{10}</math> and <math>A_{15}</math>. Using these formulas with the appropriate values will yield | |Now we have d, and we can use the same formula for <math>A_n</math> to get <math>A_{10}</math> and <math>A_{15}</math>. Using these formulas with the appropriate values will yield | ||
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|Since we found <math>A_{15}</math> in the last step, and we found the necessary pieces, we find <math>S_{10}</math> by using the formula <math>S_{10} = \frac{10}{2}(27 + -9) = 5 (-18) = -90</math> | |Since we found <math>A_{15}</math> in the last step, and we found the necessary pieces, we find <math>S_{10}</math> by using the formula <math>S_{10} = \frac{10}{2}(27 + -9) = 5 (-18) = -90</math> | ||
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A_{15} &= &27 + (-4)(15 - 1)\\ | A_{15} &= &27 + (-4)(15 - 1)\\ | ||
& =& 27 - 56\\ | & =& 27 - 56\\ | ||
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|<math>S_{10} = -90, A_{15} = -39</math> | |<math>S_{10} = -90, A_{15} = -39</math> | ||
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[[008A Sample Final A|<u>'''Return to Sample Exam</u>''']] | [[008A Sample Final A|<u>'''Return to Sample Exam</u>''']] | ||
Latest revision as of 22:54, 25 May 2015
Question: Given a sequence Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 27,23,19,15,\ldots } use formulae 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_{10}} 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 A_{15}} .
| Foundations: |
|---|
| 1) Which of the 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} formulas should you use? |
| 2) What is the common ratio or difference? |
| 3) How do you find the values you need to use the formula? |
| Answer: |
| 1) The variables in the formulae give a bit of a hint. The r stands for ratio, and ratios are associated to geometric series. This sequence is arithmetic, so we want the formula that does not involve r. |
| 2) We determine the common difference by taking two adjacent terms in the sequence, say 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} 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 A_2} , and finding their difference 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 = A_2 - A_1 = -4} |
| 3) Since we have a value for d, we want to use the 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} that involves d. |
Solution:
| Step 1: |
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| The 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 S_n} that involves a common difference, d, is the one we want. The other formula involves a common ratio, r. So we have to determine the value of n, 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} , 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 A_n} |
| Step 2: |
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| Now we determine 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} by finding d. To do this we use the formula 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)} with n = 2, 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 = 27} , andFailed 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_2 = 23} . This yields d = -4. |
| Step 3: |
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| Now we have d, and we can use the same 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} to get 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_{10}} 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 A_{15}} . Using these formulas with the appropriate values will yield |
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| and |
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| Step 4: |
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| Since we found 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_{15}} in the last step, and we found the necessary pieces, we find 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_{10}} by using the formula 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_{10} = \frac{10}{2}(27 + -9) = 5 (-18) = -90} |
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| Final Answer: |
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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 S_{10} = -90, A_{15} = -39} |