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The sequence a(1), a(2), ..., a(n), ... is such that

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The sequence a(1), a(2), ..., a(n), ... is such that [#permalink] New post 02 Mar 2012, 20:12
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The sequence a(1), a(2), …, a(n), … is such that a(n)=4a(n–1) –3 for all integers n>1. If a(3)=x, then a(1)=?

A. 4x–3
B. 16x–15
C. (x+3)/4
D. (x+3)/16
E. (x+15)/16
[Reveal] Spoiler: OA
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Re: If a(3)=x, then a(1)=? [#permalink] New post 02 Mar 2012, 20:29
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The sequence a_1, a_2, …, a_n, … is such that a_n=4a_{n-1}-3 for all integers n>1. If a_3=x, then a_1=?
A. 4x–3
B. 16x–15
C. (x+3)/4
D. (x+3)/16
E. (x+15)/16

Since, a_n=4a_{n-1}-3 then a_3=4a_{2}-3 --> x=4a_{2}-3 --> a_2=\frac{x+3}{4}.

Similarly, a_2=4a_{1}-3 --> \frac{x+3}{4}=4a_{1}-3 --> a_1=\frac{x+15}{16}.

Answer: E.

Or substitute the value for x, say x=5, then a_3=5=4a_{2}-3 --> a_2=2 --> a_2=2=4a_{1}-3 --> a_1=\frac{5}{4}. Now, just plug x=5 in the answer choices and see which one yields \frac{5}{4}: only E.

Answer: E.

Note that for plug-in method it might happen that for some particular number(s) more than one option may give "correct" answer. In this case just pick some other numbers and check again these "correct" options only. For example if you pick x=1 then you get three "correct" options A, C and E. Generally -1, 0, and 1 are not good choices for plug-in method.

Hope it helps.
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Re: If a(3)=x, then a(1)=? [#permalink] New post 02 Mar 2012, 20:34
The sequence a(1), a(2), …, a(n), … is such that a(n)=4a(n–1) –3 for all integers n>1. If a(3)=x, then a(1)=?

My Line of thought:
1. What is given to me a(3)=x, and formula.
2. I need to express a(1) from known.
3. a(n) can be expressed as a(n-1) means it can be expressed by any of the predecessor or successor.
as a(n-1) can be expressed as a(n-2)....hence a(n) can be expressed as a(n-2) and vice versa
4. a(3) = 4a(2)-3 , (n=3)>1
= 4 [4a(1)-3]-3 , (n=2)>1
=16a(1)-12-3
x = 16a(1)-15
(x+15)/16 = a(1) == Ans E
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Re: The sequence a(1), a(2), ..., a(n), ... is such that [#permalink] New post 04 Jun 2013, 04:54
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Re: The sequence a(1), a(2), ..., a(n), ... is such that [#permalink] New post 05 Jun 2013, 06:01
The sequence a(1), a(2), …, a(n), … is such that a(n)=4a(n–1) –3 for all integers n>1. If a(3)=x, then a(1)=?

A. 4x–3
B. 16x–15
C. (x+3)/4
D. (x+3)/16
E. (x+15)/16


I love such questions!

Since we know a(3) we can find a(2), if we can find a(2) we can find a(1). a(3)=4a(2)-3 ---> x=4a(2)-3 ---> a(2)=(x+3)/4

a(2)=4a(1)-3 ---> (x+3)/4=4a(1)-3 ---> 4a(1)=(x+15)/4 ---> a(1)=(x+15)/16 The answer is E.
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Re: The sequence a(1), a(2), ..., a(n), ... is such that   [#permalink] 05 Jun 2013, 06:01
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