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recursively defined

by sanju09 » Wed Jul 07, 2010 4:11 am
A sequence is recursively defined by a (n) = a (n - 1) + 2 a (n - 2), for n > 2. If a (1) = 0 and a (2) = 1, what is the value of a (6)?
(A) 5
(B) 8
(C) 11
(D) 13
(E) 21
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by Rahul@gurome » Wed Jul 07, 2010 4:34 am
a(6) = a(5) + 2a(4)

a(3) = a(2) + 2a(1) implies a(3) = 1
a(4) = a(3) + 2a(2) implies a(4) = 1 + 2 = 3
a(5) = a(4) + 2a(3) implies a(5) = 3 + 2 = 5
So, a(6) = 5 + 2(3) = 11

The correct answer is (C).
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by kmittal82 » Wed Jul 07, 2010 4:41 am
Hmm, there is probably a quicker way of doing this, but here goes:

a(6) = a(5) + 2a(4)

Now,
a(5) = a(4) + 2a(3) - eq 1
a(4) = a(3) + 2a(2) - eq 2
a(3) = a(2) + 2a(1) => a(3) = 1

Putting in eq 2
a(4) = 1 + 2(1) = 3
Putting this in eq 1
a(5) = 3 + 2(1) = 5

Finally, a(6) now becomes => 5 + 2(3) = 11

So, the answer should be (C)
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