Difference between revisions of "005 Sample Final A, Question 22"
Jump to navigation
Jump to search
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 -3, 1, -\frac{1}{3}, \frac{1}{9}, -\frac{1}{27}, \cdots }
Kayla Murray (talk | contribs) |
Kayla Murray (talk | contribs) |
||
| Line 7: | Line 7: | ||
! Step 1: | ! Step 1: | ||
|- | |- | ||
| − | | | + | | The sequence is a geometric sequence. The common ratio is <math>r=\frac{-1}{3}</math>. |
|} | |} | ||
| Line 13: | Line 13: | ||
! Step 2: | ! Step 2: | ||
|- | |- | ||
| − | | | + | | The formula for the nth term of a geometric series is <math>a_n=ar^{n-1}</math> where <math>a</math> is the first term of the sequence. |
| + | |- | ||
| + | | So, the formula for this geometric series is <math>a_n=(-3)\left(\frac{-1}{3}\right)^{n-1}</math>. | ||
|} | |} | ||
| Line 19: | Line 21: | ||
! Step 3: | ! Step 3: | ||
|- | |- | ||
| − | | | + | | For geometric series, <math>\displaystyle{\sum_{k=1}^\infty a_k}=\frac{a}{1-r}</math> if <math>|r|<1</math>. Since <math>|r|=\frac{1}{3}</math>, |
|- | |- | ||
| − | | | + | | we have <math>\displaystyle{\sum_{k=1}^\infty a_k}=\frac{-3}{1-\frac{-1}{3}}=\frac{-9}{4}</math>. |
|} | |} | ||
{| class="mw-collapsible mw-collapsed" style = "text-align:left;" | {| class="mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| − | ! | + | ! Final Answer: |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
|- | |- | ||
| − | | | + | | <math>a_n=(-3)\left(\frac{-1}{3}\right)^{n-1}</math> |
| − | |||
| − | |||
| − | |||
| − | |||
| − | { | ||
| − | |||
|- | |- | ||
| − | | | + | |<math>\frac{-9}{4}</math> |
|} | |} | ||
Revision as of 12:03, 20 May 2015
Question Consider the following sequence,
a. Determine a 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}
, the n-th term of the sequence.
b. Find the sum 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_{k=1}^\infty a_k}}
| Step 1: |
|---|
| The sequence is a geometric sequence. The common ratio 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 r=\frac{-1}{3}} . |
| Step 2: |
|---|
| The formula for the nth term of a geometric series 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=ar^{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} is the first term of the sequence. |
| So, the formula for this geometric series 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=(-3)\left(\frac{-1}{3}\right)^{n-1}} . |
| Step 3: |
|---|
| For geometric 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 \displaystyle{\sum_{k=1}^\infty a_k}=\frac{a}{1-r}} if 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 |r|<1} . 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 |r|=\frac{1}{3}} , |
| 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 \displaystyle{\sum_{k=1}^\infty a_k}=\frac{-3}{1-\frac{-1}{3}}=\frac{-9}{4}} . |
| 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 a_n=(-3)\left(\frac{-1}{3}\right)^{n-1}} |
| 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{-9}{4}} |