We Define F0 1 Then F is Continuous and I is Not Improper
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Define a sequence recursively by F0 = 0, F1=1, and Fn= the remainder [#permalink] 
09 May 2019, 23:40
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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
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Re: Define a sequence recursively by F0 = 0, F1=1, and Fn= the remainder [#permalink] 10 May 2019, 00:52
Bunuel wrote:
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
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)
• …..
So, the pattern is 0, 1, 1, 2, 0, 2, 2, 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)
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Re: Define a sequence recursively by F0 = 0, F1=1, and Fn= the remainder [#permalink] 18 May 2022, 23:19
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Re: Define a sequence recursively by F0 = 0, F1=1, and Fn= the remainder [#permalink]
18 May 2022, 23:19
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