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The infinite sequence \(a_{1}\), \(a_{2}\), …, \(a_{n}\), … is such that \(a_{1} = 7\), \(a_{2} = 8\), \(a_{3} = 10\), and \(a_n=a_{n-3} + 7\) for values of n > 3. What is the remainder when \(a_{n}\) is divided by 7?
(1) n is a multiple of 3.
(2) n is an even number.
(1) n is a multiple of 3.
(2) n is an even number.
Disclaimer: I have used the Search Box Before Posting. I used the first sentence of the question or a string of words exactly as they show up in the question below for my search. I did not receive an exact match for my question.
Source: Veritas Prep; Book 04
Chapter: Homework
Topic: Algebra
Question: 94
Question: Page 227
Edition: Third
My Question: Please provide an explanation on how to arrive at the answer.
Source: Veritas Prep; Book 04
Chapter: Homework
Topic: Algebra
Question: 94
Question: Page 227
Edition: Third
My Question: Please provide an explanation on how to arrive at the answer.
26 Jul 2013, 12:40
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The infinite sequence \(a_{1}\), \(a_{2}\), …, \(a_{n}\), … is such that \(a_{1} = 7\), \(a_{2} = 8\), \(a_{3} = 10\), and \(a_n=a_{n-3} + 7\) for values of n > 3. What is the remainder when \(a_{n}\) is divided by 7?
\(a_{1} = 7\)
\(a_{2} = 8\)
\(a_{3} = 10\)
\(a_{4} = a_1+7=7+7=14\)
\(a_{5} = a_2+7=8+7=15\)
\(a_{6} = a_3+7=10+7=17\)
...
Notice that the remainder upon division the above terms by 7 repeats in blocks of 3: {0, 1, 3} {0, 1, 3}...
(1) n is a multiple of 3 --> every third term has the remainder of 3 (\(a_{3}\), \(a_{6}\), \(a_{9}\), ...). Sufficient.
(2) n is an even number. Not sufficient: consider \(a_2\) and \(a_4\).
Answer: A.
\(a_{1} = 7\)
\(a_{2} = 8\)
\(a_{3} = 10\)
\(a_{4} = a_1+7=7+7=14\)
\(a_{5} = a_2+7=8+7=15\)
\(a_{6} = a_3+7=10+7=17\)
...
Notice that the remainder upon division the above terms by 7 repeats in blocks of 3: {0, 1, 3} {0, 1, 3}...
(1) n is a multiple of 3 --> every third term has the remainder of 3 (\(a_{3}\), \(a_{6}\), \(a_{9}\), ...). Sufficient.
(2) n is an even number. Not sufficient: consider \(a_2\) and \(a_4\).
Answer: A.
26 Jul 2013, 12:42
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