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.

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  • 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.
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  • Follow example below.
  • Click here to refer to solution format in 7.7



Question 1
What's the interval over which this infinite series converges?


Solution 1


4 < x - 3
x > 7

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

Error Explanation 1
A) x > -7 or x < -1
B) x > -7
C) x < -1
D) -1 < x < 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, ... ?

Solution 2



Answer: B

Error Explanation 2
A)

B)

C)

D)


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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Question 3
Given the rule of:

Find the sum of the first 11 terms (calc. okay).

Solution 3
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.
Answer: B

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

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

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

The formula



seems the appropriate choice, but let’s check.
Thus, all is well and the solution is





Error Explanation 4
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.
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Question 5
Find the 4th term of the recursive sequence

where a1=4

Solution 5

a1 = 4, gives




Error Explanation 5
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
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Question 6
Compute the sum


Solution 6



Error Explanation 6
a) added wrong
b) added wrong
c) correct
d) denominator is correct, but numerator added wrong
e) c is answer, so this is wrong
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Question 7
Compute the sum



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





Error Explanation 7
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...
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Question 8
Find the sum of the infinite series



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





Error Explanation 8
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
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Question 9
Solution 9





Error Explanation 9
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.

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Question 10
Solution 10
Because
S = 3 + 5 + 7 + · · · + 101,
we have a = 3 and d = 2, so



Use an = 101 to determine the number of terms.

->1 + 2n = 101
->2n = 100
->n = 50
Thus, there are 50 terms. We can now use



S = 2600 (B)
Error Explanation 10
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.

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Question 11
Solution 11



(A)
Error Explanation 11
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.

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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
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Question 25