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10 May 2019, 00:40
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
68% (03:07) correct
32%
(02:41)
wrong
based on 288
sessions
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Not Attempted Yet
Define a sequence recursively by \(F_{0}=0\), \(F_{1}=1\), and \(F_{n}=\) the remainder when \(F_{n-1}+F_{n-2}\) is divided by 3, for all \(n\geq 2\). Thus the sequence starts 0, 1, 1, 2, 0, 2, .... What is \(F_{2017}+F_{2018}+F_{2019}+F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}\)?
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
10 May 2019, 01:52
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Bunuel
The pattern in this case is as shown
- •\(0 – F_0\)
• \(1 – F_1\)
• \(1 – F_2\) (0+1/3 -> remainder is 1)
• \(2 – F_3\) (1+1/3 -> remainder is 2)
• \(0 – F_4\) (1+2/3 -> remainder is 0)
• \(2 – F_5\) (2+0/3 -> remainder is 2)
• \(2 – F_6\) (0+2/3 -> remainder is 2)
• \(1 - F_7\) (2+2/3 -> remainder is 1)
• \(0 – F_8\) (2+1/3 -> remainder is 0) {Note that the pattern again repeats itself from here}
• \(1 – F_9\) (1+0/3 -> remainder is 1)
• \(1 – F_{10}\) (0+1/3 -> remainder is 1)
• …..
The pattern starts again at \(F_8\) and continue after every 8 terms, so \(F_{16},F_{24}\) and so on marks the start of the pattern.
We are starting from \(F_{2017}\), now 2017 divided by 8 gives us remainder 1.
That means \(F_{2016}\) was one of the starting points and must have been 0.
• \(F_{2017} = 1\)
• \(F_{2018} = 1\)
• \(F_{2019} = 2\)
• \(F_{2020} = 0\)
• \(F_{2021} = 2\)
• \(F_{2022} = 2\)
• \(F_{2023} = 1\)
• \(F_{2024} = 0\)
Sum will be equal to 9. (Option D)
12 Sep 2024, 07:45
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\(F_{0}=0\), \(F_{1}=1\), and \(F_{n}=\) the remainder when \(F_{n-1}+F_{n-2}\) is divided by 3, for all \(n\geq 2\) and we need to find the value of \(F_{2017}+F_{2018}+F_{2019}+F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}\)
=> \(F_{2} \) = Remainder of \( ( F_{0} + F_{1}) \) by 3 = Remainder of (0 + 1) by 3 = Remainder of 1 by 3
Similarly, \(F_{3}\) = Remainder of (1 + 1) by 3 = 2
\(F_{4}\) = Remainder of (1 + 2) by 3 = 0
\(F_{5}\) = Remainder of (2 + 0) by 3 = 2
\(F_{6}\) = Remainder of (0 + 2) by 3 = 2
\(F_{7}\) = Remainder of (2 + 2) by 3 = 1
\(F_{8}\) = Remainder of (2 + 1) by 3 = 0
\(F_{9}\) = Remainder of (1 + 0) by 3 = 1
\(F_{10}\) = Remainder of (0 + 1) by 3 = 1
\(F_{11}\) = Remainder of (1 + 1) by 3 = 2
So, series is 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2
=> Series is repeating after a set of 8 values
2017 = 2016 + 1 = Multiple of 8 + 1
=> \(F_{2017}\) = 0 [ First term after the set of 8 terms ]
=> \(F_{2018}\) = 1 [ First term after the set of 8 terms ]
=> \(F_{2019}\) = 1 [ First term after the set of 8 terms ]
=> \(F_{2020}\) = 2 [ First term after the set of 8 terms ]
=> \(F_{2021}\) = 0 [ First term after the set of 8 terms ]
=> \(F_{2022}\) = 2 [ First term after the set of 8 terms ]
=> \(F_{2023}\) = 2 [ First term after the set of 8 terms ]
=> \(F_{2024}\) = 1 [ First term after the set of 8 terms ]
=> \(F_{2017}+F_{2018}+F_{2019}+F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}\) = 0 + 1 + 1 + 2 + 0 + 2 + 2 + 1 = 9
So, Answer will be D
Hope it helps!
Watch the following video to MASTER Sequences
=> \(F_{2} \) = Remainder of \( ( F_{0} + F_{1}) \) by 3 = Remainder of (0 + 1) by 3 = Remainder of 1 by 3
Similarly, \(F_{3}\) = Remainder of (1 + 1) by 3 = 2
\(F_{4}\) = Remainder of (1 + 2) by 3 = 0
\(F_{5}\) = Remainder of (2 + 0) by 3 = 2
\(F_{6}\) = Remainder of (0 + 2) by 3 = 2
\(F_{7}\) = Remainder of (2 + 2) by 3 = 1
\(F_{8}\) = Remainder of (2 + 1) by 3 = 0
\(F_{9}\) = Remainder of (1 + 0) by 3 = 1
\(F_{10}\) = Remainder of (0 + 1) by 3 = 1
\(F_{11}\) = Remainder of (1 + 1) by 3 = 2
So, series is 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2
=> Series is repeating after a set of 8 values
2017 = 2016 + 1 = Multiple of 8 + 1
=> \(F_{2017}\) = 0 [ First term after the set of 8 terms ]
=> \(F_{2018}\) = 1 [ First term after the set of 8 terms ]
=> \(F_{2019}\) = 1 [ First term after the set of 8 terms ]
=> \(F_{2020}\) = 2 [ First term after the set of 8 terms ]
=> \(F_{2021}\) = 0 [ First term after the set of 8 terms ]
=> \(F_{2022}\) = 2 [ First term after the set of 8 terms ]
=> \(F_{2023}\) = 2 [ First term after the set of 8 terms ]
=> \(F_{2024}\) = 1 [ First term after the set of 8 terms ]
=> \(F_{2017}+F_{2018}+F_{2019}+F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}\) = 0 + 1 + 1 + 2 + 0 + 2 + 2 + 1 = 9
So, Answer will be D
Hope it helps!
Watch the following video to MASTER Sequences
