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a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divi

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by Max@Math Revolution » Fri Feb 08, 2019 1:24 am

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[GMAT math practice question]

a0 = 0, a1 = 1. an is the remainder when an-1 + an-2 is divided by 3 for n ≥ 2. What is the value of a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108?

A. 0
B. 3
C. 5
D. 7
E. 9
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by Max@Math Revolution » Sun Feb 10, 2019 4:55 pm
=>

a0 = 0
a1 = 1
a2 = a1 + a0 = 1 + 0 = 1
a3 = a_2 + a1 = 1 + 1 = 2
a4 = 0 since a3 + a2 = 2 + 1 = 3 = 3(1)+0, and the remainder when a3 + a2 is divided by 3 is zero.
a5 = a4 + a3 = 0 + 2 = 2
a6 = a5 + a4 = 2 + 0 = 2
a7 = 1 since a6 + a5 = 2 + 2 = 4= 3(1)+1, and the remainder when a6 + a5 is divided by 3 is 1.
a8 = 0 since a7 + a6 = 1 + 2 = 3= 3(1)+0, the remainder is 0, when a7 + a6 is divided by 3.
a9 = a8 + a7 = 0 + 1 = 1
Thus, the sequence is periodic, with period 8.
a101 + a102 + a103 + a104 + a105 + a106 + a107 + a108
= a5 + a6 + a7 + a0 + a1 + a2 + a3 + a4
= 2 + 2 + 1 + 0 + 1 + 1 + 2 + 0 = 9
Therefore, the answer is E.
Answer: E
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