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Updated on: 12 Apr 2026, 10:06
Originally posted by bhanuvemula on 29 Mar 2007, 18:53.
Last edited by Bunuel on 12 Apr 2026, 10:06, edited 2 times in total.
Last edited by Bunuel on 12 Apr 2026, 10:06, edited 2 times in total.
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60% (02:45) correct
40%
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A sequence \(a_1=64\), \(a_2=66\), \(a_3=67\), and \(a_n=8+a_{(n-3)}\), which of the following is in the sequence?
A. 105
B. 786
C. 966
D. 1025
E. 1076
A. 105
B. 786
C. 966
D. 1025
E. 1076
19 Jan 2011, 20:30
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A sequence, a1=64, a2=66, a3=67, an=8+an-3, which of the following is in the sequence?
105
786
966
1025
Given:
\(a_1 = 64\)
\(a_2 = 66\)
\(a_3 = 67\)
...
\(a_n = 8 + a_{n - 3}\)
So \(a_4 = 8 + a_1\) = 8 + 64
\(a_5 = 8 + a_2\) = 8 + 66
\(a_6 = 8 + a_3\) = 8 + 67
\(a_7 = 8 + a_4\) = 8 + 8 + 64
\(a_8 = 8 + a_5\) = 8 + 8 + 66
and so on...
So any number that belongs to this sequence will be sum of one of 64/66/67 and some number of 8s.
105 - 64 = 41 which is not a multiple of 8. 41 is 1 more than a multiple of 8 so when you subtract 2/3 out of 41 (in effect subtracting 66/67 out of 105), you will still not get a multiple of 8. Hence 105 is not in this sequence.
786 - 64 = 720 which is divisible by 8 hence it will be in the sequence. This is your answer and ideally you should stop here. But if you want to check the remaining two options:
966 - 64 = 902 which is not divisible by 8. Neither are 900 and 899. Or say that 902 is 6 more than a multiple of 8 so when you subtract 2/3 out of it, you will still not get a multiple of 8.
1025 - 64 = 961 which is not divisible by 8 and is 1 more than a multiple of 8 so when you subtract 2/3 out of it, it will still not give a multiple of 8.
105
786
966
1025
Given:
\(a_1 = 64\)
\(a_2 = 66\)
\(a_3 = 67\)
...
\(a_n = 8 + a_{n - 3}\)
So \(a_4 = 8 + a_1\) = 8 + 64
\(a_5 = 8 + a_2\) = 8 + 66
\(a_6 = 8 + a_3\) = 8 + 67
\(a_7 = 8 + a_4\) = 8 + 8 + 64
\(a_8 = 8 + a_5\) = 8 + 8 + 66
and so on...
So any number that belongs to this sequence will be sum of one of 64/66/67 and some number of 8s.
105 - 64 = 41 which is not a multiple of 8. 41 is 1 more than a multiple of 8 so when you subtract 2/3 out of 41 (in effect subtracting 66/67 out of 105), you will still not get a multiple of 8. Hence 105 is not in this sequence.
786 - 64 = 720 which is divisible by 8 hence it will be in the sequence. This is your answer and ideally you should stop here. But if you want to check the remaining two options:
966 - 64 = 902 which is not divisible by 8. Neither are 900 and 899. Or say that 902 is 6 more than a multiple of 8 so when you subtract 2/3 out of it, you will still not get a multiple of 8.
1025 - 64 = 961 which is not divisible by 8 and is 1 more than a multiple of 8 so when you subtract 2/3 out of it, it will still not give a multiple of 8.
A sequence \(a_1=64\), \(a_2=66\), \(a_3=67\), and \(a_n=8+a_{(n-3)}\), which of the following is in the sequence?
A. 105
B. 786
C. 966
D. 1025
E. 1076
Easier way would be to write down several terms from the sequence:
\(a_1 = 64\)
\(a_2 = 66\)
\(a_3 = 67\)
\(a_4 = 8 + a_1 = 72\)
\(a_5 = 8 + a_2 = 74\)
\(a_6 = 8 + a_3 = 75\)
...
\(a_n = 8 + a_{n - 3}\)
Note that the terms in the sequence have remainder of 0 (\(a_1\), \(a_4\), \(a_7\), ...), 2 (\(a_2\), \(a_5\), \(a_8\), ...) or 3 (\(a_3\), \(a_6\), \(a_9\), ...) upon division by 8. Only 786 has appropriate remainder of 2 (105 and 1025 has a remainder of 1 upon division by 8, 966 has a remainder of 6, 1076 has a remainder of 4).
Answer: B.
Hope it's clear.
A. 105
B. 786
C. 966
D. 1025
E. 1076
Easier way would be to write down several terms from the sequence:
\(a_1 = 64\)
\(a_2 = 66\)
\(a_3 = 67\)
\(a_4 = 8 + a_1 = 72\)
\(a_5 = 8 + a_2 = 74\)
\(a_6 = 8 + a_3 = 75\)
...
\(a_n = 8 + a_{n - 3}\)
Note that the terms in the sequence have remainder of 0 (\(a_1\), \(a_4\), \(a_7\), ...), 2 (\(a_2\), \(a_5\), \(a_8\), ...) or 3 (\(a_3\), \(a_6\), \(a_9\), ...) upon division by 8. Only 786 has appropriate remainder of 2 (105 and 1025 has a remainder of 1 upon division by 8, 966 has a remainder of 6, 1076 has a remainder of 4).
Answer: B.
Hope it's clear.
Bhanu
