Difference between revisions of "008A Sample Final A, Question 14"
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(Created page with "'''Question: ''' Compute <math> \displaystyle{\sum_{n=1}^\infty 5\left(\frac{3}{5}\right)^n}</math> {| class="mw-collapsible mw-collapsed" style = "text-align:left;" !Founda...") |
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| − | !Foundations | + | !Foundations: |
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|1) What type of series is this? | |1) What type of series is this? | ||
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| − | ! Step 1: | + | !Step 1: |
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|We start by identifying this series as a geometric series, and the desired formula for the sum being <math> S_\infty = \frac{a_1}{1 - r}</math>. | |We start by identifying this series as a geometric series, and the desired formula for the sum being <math> S_\infty = \frac{a_1}{1 - r}</math>. | ||
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| − | ! Step 2: | + | !Step 2: |
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|Since <math>a_1</math> is the first term in the series, <math>a_1 = 5\frac{3}{5} = 3</math>. The value for r is the ratio between consecutive terms, which is <math>\frac{3}{5}</math>. Plugging everything in we have <math> S_\infty = \frac{3}{1-\frac{3}{5}} = \frac{3}{\frac{2}{5}} = \frac{15}{2}</math> | |Since <math>a_1</math> is the first term in the series, <math>a_1 = 5\frac{3}{5} = 3</math>. The value for r is the ratio between consecutive terms, which is <math>\frac{3}{5}</math>. Plugging everything in we have <math> S_\infty = \frac{3}{1-\frac{3}{5}} = \frac{3}{\frac{2}{5}} = \frac{15}{2}</math> | ||
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| − | ! Final Answer: | + | !Final Answer: |
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|<math>\frac{15}{2}</math> | |<math>\frac{15}{2}</math> | ||
Latest revision as of 23:01, 25 May 2015
Question: 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 \displaystyle{\sum_{n=1}^\infty 5\left(\frac{3}{5}\right)^n}}
| Foundations: |
|---|
| 1) What type of series is this? |
| 2) Which formula, on the back page of the exam, is relevant to this question? |
| 3) In the formula there are some placeholder variables. What is the value of each placeholder? |
| Answer: |
| 1) This series is geometric. The giveaway is there is a number raised to the nth power. |
| 2) The desired formula 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_\infty = \frac{a_1}{1-r}} |
| 3) 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 in the series, which 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 5\frac{3}{5} = 3} . The value for r is the ratio between consecutive terms, which 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 \frac{3}{5}} |
Solution:
| Step 1: |
|---|
| We start by identifying this series as a geometric series, and the desired formula for the sum being 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_\infty = \frac{a_1}{1 - r}} . |
| Step 2: |
|---|
| 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 a_1} is the first term in the 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 a_1 = 5\frac{3}{5} = 3} . The value for r is the ratio between consecutive terms, which 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 \frac{3}{5}} . Plugging everything in 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 S_\infty = \frac{3}{1-\frac{3}{5}} = \frac{3}{\frac{2}{5}} = \frac{15}{2}} |
| 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 \frac{15}{2}} |