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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|<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 23:54, 25 May 2015

Question: Given a sequence use formulae to compute and .

Foundations:  
1) Which of the 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 and , and finding their difference
3) Since we have a value for d, we want to use the formula for that involves d.

Solution:

Step 1:  
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, , and
Step 2:  
Now we determine by finding d. To do this we use the formula with n = 2, , and. This yields d = -4.
Step 3:  
Now we have d, and we can use the same formula for to get and . Using these formulas with the appropriate values will yield
and
Step 4:  
Since we found in the last step, and we found the necessary pieces, we find by using the formula
Final Answer:  

Return to Sample Exam