12.7+OFSA+Solutions


 * OFSA SOLUTIONS for Sequences and Series**

This page contains peer generated solutions and error explanations to OFSA questions. As you read or view the solutions, be critical: check for accuracy, but also for more efficient solution strategies. If you have a better method or different idea/answer, post a discussion and monitor the responses.


 * Quick Directions**
 * Post answers, solutions and error explanations to each OFSA question below.
 * For each "distractor" or incorrect answer choice, explain the error that would lead to that incorrect answer choice.
 * You may either do the above in typed format or using a pencast.
 * Separate each question with a section bar.
 * After each solution, provide a hyperlink back to the corresponding OFSA page.
 * Follow example below.
 * Click here to refer to solution format in 7.7

math \frac{1}{x}+\frac{4}{x(x-3)}+\frac{16}{x(x-3)^{2}}+\frac{64}{x(x-3)^{3}}+... math
 * Question 1**
 * What's the interval over which this infinite series converges?**

math math math -1<\frac{4}{x-3}<1 math 4 < x - 3 x > 7
 * __Solution 1__**
 * r|<1

x - 3 < -4 x < -1
 * Answer: A**

A) x > -7 or x < -1 B) x > -7 C) x < -1 D) -1 < x < 1
 * __Error Explanation 1__**

A: Correct B: It's not just greater than 7; must check other side. C: It's not just less than -1; must check other side. D: This is just the concept of the magnitude of the ratio //r// must be between -1 and 1.

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 * Question 2**
 * What's the explicit geometric rule for 10, -5, 2.5, -1.25, ... ?**

math r=\frac{-5}{10}=\frac{-1}{2} math math a_{1}=10 math math a_{n}=10(-\frac{1}{2})^{n-1} math
 * __Solution 2__**
 * Answer: B**

A) math a_{n}=10(\frac{1}{2})^{n-1} math B) math a_{n}=10(\frac{-1}{2})^{n-1} math C) math a_{n}=10(\frac{1}{2})^{n} math D) math a_{n}=10(2)^{n-1} math
 * __Error Explanation 2__**

A: This one doesn't have a negative in front of the ratio. B: Correct C: This one doesn't have the exponent of (n-1). If this is true, it would start at n=0 for the first term to be 10. D: The ratio is wrong in this choice.

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math a_{n}=3^{n}+(-2)^{n} math
 * Question 3**
 * Given the rule of:**
 * Find the sum of the first 11 terms (calc. okay).**

Type in your calculator: TI 83/84 - sum(seq(3^x+(-2)^x, x, 1, 11)) TI 89 - F3, 4, then S (3^x+(-2)^x, x, 1, 11) You get out the answer 264,353.
 * __Solution 3__**
 * Answer: B**

A) 175,009 B) 264,353 C) 2,049 D) 125,635
 * __Error Explanation 3__**

A: When you just plug in 11 for the rule, you get this number, which is just the 11th term. B: Correct C: You get this answer if you try to use a partial sum of a geometric sequence rule, by using the first term as 3, the ratio as -2, and the number of terms as 11. D: Random answer.

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1, -8, 27, -64...
 * Question 4**
 * Find the nth term of the sequence**

Note that each term is the cube of a natural number and that the signs alternated.
 * __Solution 4__**

The formula

math a_{n} = (-1)^{n+1}*n^{3} math

seems the appropriate choice, but let’s check. Thus, all is well and the solution is math a1 = (-1)^{1+1}*(1)^{3} = (1)*(1) = 1 math math a2 = (-1)^{2+1}*(2)^{3} = (−1)*(8) = -8 math math a3 = (-1)^{3+1}*(3)^{3} = (1)*(27) = 27 math math a4 = (-1)^{4+1}*(4)^{3} = (−1)*(64) = -64 math math a_{n} = (-1)^{n+1}*n^{3}. math a) sign of even number of n is incorrect b) sign of odd number of n is incorrect c) correct d) it will not work after first one because it is not square. e) same as d) that it will not work on after first one because it is not square. Click here to return to OFSA.
 * __Error Explanation 4__**

math a_{n}=3*a_{n-1}+2 math
 * Question 5**
 * Find the 4th term of the recursive sequence**
 * where a1=4**


 * __Solution 5__**

a1 = 4, gives math a_{2} = (3)a_{1} + 2 = 3(4) + 2 = 14 math math a_{3} = (3)a_{2} + 2 = 3(14) + 2 = 44 math math a_{4} = (3)a_{3} + 2 = 3(44) + 2 = 134. math

a) answer is 134 from above, so incorrect b) answer is 134 from above, so incorrect c) answer is 134 from above, so incorrect d) answer is 134 from above, so incorrect e) correct because none of answers are correct Click here to return to OFSA.
 * __Error Explanation 5__**

math \sum_{3}^{k=1} \frac{k}{k+1} math
 * Question 6**
 * Compute the sum**

math \frac{1}{2}+\frac{2}{3}+\frac{3}{4} math math =\frac{6}{12}+\frac{8}{12}+\frac{9}{12} math math =\frac{23}{12} math a) added wrong b) added wrong c) correct d) denominator is correct, but numerator added wrong e) c is answer, so this is wrong Click here to return to OFSA.
 * __Solution 6__**
 * __Error Explanation 6__**


 * Question 7**
 * Compute the sum**

math 1-x+x^{2}-x^{3}+x^{4}-x^{5} math

This is geometric with common ration r = -x. Thus, the sum of these 6 terms is
 * __Solution 7__**

math \[S_{n}= \frac{a(1-r^{n})}{1-r}\] math math \[S_{6}= \frac{1(10(-x)^{6})}{1+x}\] math math \[S_{6}= \frac{1-x^{6}}{1+x}\] math

a) denominator is correct, but numerator added wrong b) sign and numerator added wrong c) sign of numerator is not +x^6 d) correct e) There is no a in this question... Click here to return to OFSA.
 * __Error Explanation 7__**


 * Question 8**
 * Find the sum of the infinite series**

math \sum_{n=0}^{\infty} \frac{3}{2^{n}} math

This is geometric with common ration r = 1/2. Because −1 < r < 1, the series converges and
 * __Solution 8__**

math \[S= \frac{a}{1-r}\] math math \[S= \frac{3}{1-1/2}\] math math S= 6 math

a) forget about denominator, so incorrect b) add 1 to a)`s answer, but still incorrect c) correct d) added denominator instead of subtract them(1+r) e) answer is c, so this is incorrect Click here to return to OFSA.
 * __Error Explanation 8__**

math \[\textup{{The Sequence 1,\, 5,\, 9,\, 13} \, \: is\: arithmetic,\: with\: common\: ratio\: d = 4, so:}\] math math \[\rightarrow a_{n}= a_{1}+(n-1)d\] math math \[\rightarrow a_{n}= 1+(n-1)4\] math math \[\rightarrow a_{200}= 1-(200-1)4= 797\] math math \[\therefore a_{200}= \textbf{797}\] (A) math B) Must remember to subtract 1 from the number of terms to get the right answer. C) Important to remember that when using rule to subtract 1 from 200 not add. D) Be sure to use the correct distance between consecutive terms.
 * Question 9**
 * __Solution 9__**
 * __Error Explanation 9__**

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Because S = 3 + 5 + 7 + · · · + 101, we have a = 3 and d = 2, so math \[\rightarrow a_{n}= a_{1}+(n-1)d\] math math \[\rightarrow a_{n}= 3+(n-1)2\] math math \[\rightarrow a_{n}= 1+2n math Use an = 101 to determine the number of terms. math \[\rightarrow a_{n}=101 math ->1 + 2n = 101 ->2n = 100 ->n = 50 Thus, there are 50 terms. We can now use
 * Question 10**
 * __Solution 10__**

math \[\rightarrow S=\frac{n\left ( a_{1}+a_{n} \right )}{2}\] math math \[\rightarrow S=\frac{50\left ( 3+101\right )}{2}\] math S = 2600 (B) A) Make sure to use correct distance between terms. B) ~ C) Make sure to find the number of terms located in the sequence first. D) Make sure to find the number of terms located in the sequence first.
 * __Error Explanation 10__**

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math \[\rightarrow S_{n}=\frac{a\left (1-r^{n})}{1-r}\] math math \[\rightarrow S_{10}=\frac{1\left (1-3^{10})}{1-3}\] math math \[\rightarrow S_{10}=29,524 math A) ~ B) Make sure you are using the correct equation and remember it is geometric. C) Make sure not to add the ration when calculating the sum. D) Place corrects signs in each area of the equation as to not mess you up.
 * Question 11**
 * __Solution 11__**
 * __(A)__**
 * __Error Explanation 11__**

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 * Question 12**
 * __Solution 12__**
 * __Error Explanation 12__**

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 * Question 13**
 * __Solution 13__**
 * __Error Explanation 13__**

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 * Question 14**
 * __Solution 14__**
 * __Error Explanation 14__**

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 * Question 15**
 * __Solution 15__**
 * __Error Explanation 15__**

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 * Question 16**
 * __Solution 16__**
 * __Error Explanation 16__**

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 * Question 17**
 * __Solution 17__**
 * __Error Explanation 17__**

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 * Question 18**
 * __Solution 18__**
 * __Error Explanation 18__**

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 * Question 19**
 * __Solution 19__**
 * __Error Explanation 19__**

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 * Question 20**
 * __Solution 20__**
 * __Error Explanation 20__**

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 * Question 21**
 * __Solution 21__**
 * __Error Explanation 21__**

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 * Question 22**
 * __Solution 22__**
 * __Error Explanation 22__**

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 * Question 23**
 * __Solution 23__**
 * __Error Explanation 23__**

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 * Question 24**
 * __Solution 24__**
 * __Error Explanation 24__**


 * Question 25**