If the sequence {An } satisfies An = An-1 - An-2, A1 = 0, an

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[GMAT math practice question]

If the sequence {An } satisfies An = An-1 - An-2, A1 = 0, and A2 = 1, where n is an integer greater than 2, then what is the sum of the first 100 terms of {An }?

A. 1
B. 2
C. 3
D. 4
E. 5

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by GMATGuruNY » Thu Jul 12, 2018 2:18 am
Max@Math Revolution wrote:[GMAT math practice question]

If the sequence {An } satisfies An = An-1 - An-2, A1 = 0, and A2 = 1, where n is an integer greater than 2, then what is the sum of the first 100 terms of {An }?

A. 1
B. 2
C. 3
D. 4
E. 5
A� = 0.
Aâ‚‚ = 1.
A₃ = A₂ - A� = 1 - 0 = 1.
A₄ = A₃ - A₂ = 1 - 1 = 0.
A₅ = A₄ - A₃ = 0 - 1 = -1.
A₆ = A₅ - A₄ = -1 - 0 = -1.
A₇ = A₆ - A₅ = -1 - (-1) = 0.
A₈ = A₇ - A₆ = 0 - (-1) = 1.

The alternating blue and red implies the following pattern:
0, 1, 1...0, -1, -1...0, 1, 1...0, -1. -1...

The sum of every blue-red cycle = 0 + 1 + 1 + 0 + (-1) + (-1) = 0.
Since every blue-red cycle is composed of 6 terms -- and 96 = 16*6 -- the first 96 terms will be composed of 16 blue-red cycles, yielding a sum of 0.
The 97th through 100th terms will be composed of 3 blue terms and 1 red term, yielding the following sum:
0 + 1 + 1 + 0 = 2.

The correct answer is B.
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by Max@Math Revolution » Sun Jul 15, 2018 5:05 pm
=>

We determine the period of the sequence by examining its terms:
A1 = 0 and A2 = 1.
A3 = A2 - A1 = 1 - 0 = 1
A4 = A3 - A2 = 1 - 1 = 0
A5 = A4 - A3 = 0 - 1 = -1
A6 = A5 - A4 = -1 - 0 = -1
A7 = A6 - A5 = -1 - (-1) = 0
A8 = A7 - A6 = 0 - (-1) = 1
A9 = A8 - A7 = 1 - 0 = 1
A10 = A9 - A8 = 1 - 1 = 0
...

The sequence has period 6.
The sum of the first six terms is A1 + A2 + ... A6 = 0 + 1 + 1 + 0 + (-1) + (-1) = 0.
We apply this fact to determine the sum of the first 100 terms of the sequence:
A1 + A2 + ... A100
= ( A1 + A2 + ... A6 ) + ... + ( A91 + A92 + ... A96 ) + A97 + A98 + A99 + A100
= 0 + ... + 0 + A97 + A98 + A99 + A100
= A97 + A98 + A99 + A100
= 0 + 1 + 1 + 0 = 2.

Therefore, the answer is B.

Answer : B

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by Scott@TargetTestPrep » Thu Jul 19, 2018 11:41 am
Max@Math Revolution wrote:[GMAT math practice question]

If the sequence {An } satisfies An = An-1 - An-2, A1 = 0, and A2 = 1, where n is an integer greater than 2, then what is the sum of the first 100 terms of {An }?

A. 1
B. 2
C. 3
D. 4
E. 5
Let's list the first few terms to discern a pattern.

A1 = 0
A2 = 1
A3 = 1 - 0 = 1
A4 = 1 - 1 = 0
A5 = 0 - 1 = -1
A6 = -1 - 0 = -1
A7 = -1 - (-1) = 0
A8 = 0 - (-1) = 1
A9 = 1 - 0 = 1

At this point, we can see that the terms repeat themselves in a cycle of 6 numbers: 0, 1, 1, 0, -1, -1 (notice that A7 = A1, A8 = A2, A9 = A3, etc.). Also notice that the sum of the 6 numbers in one cycle is 0. So the sum of all the terms up to and including the 96th term is 0 (notice 96 = 6 x 16). So we really just need to add A97, A98, A99 and A 100. Since A97 = A1 = 0, A98 = A2 = 1, A99 = 1 and A100 = 0, the sum of these 4 terms (and hence the sum of the first 100 terms) is 0 + 1 + 1 + 0 = 2.

Answer: B

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