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If the sequence...

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by alex.gellatly » Tue Jun 05, 2012 5:55 am
If the sequence x<1>, x<2>,.....x<n>, is such that x<1> = 3 and x<n+1> = 2x<n>-1 for n greater than or equal to 1, then x<20>-x<19>=

A. 2^19
B. 2^20
C 2^21
D 2^20 - 1
E 2^21 - 1

Thanks
Last edited by alex.gellatly on Sun Jun 10, 2012 12:04 am, edited 1 time in total.
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by 1947 » Tue Jun 05, 2012 7:32 am
not a clue :(
If my post helped you- let me know by pushing the thanks button. Thanks
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by eagleeye » Tue Jun 05, 2012 7:53 am
The correct answer should be [spoiler]2^19[/spoiler]. Let me explain:

We'll do this by pattern recognition.

We are told that X(1)=3

Also we are given that X(n+1)=2X(n)-1

Let's find the first few terms.
X(2) = 2*X(1)-1 = 2*3-1 = 5
X(3) = 2*X(2)-1 = 2*5-1 = 9
X(4) = 2*X(3)-1= 2*9-1 = 17

We need to find X(20)-X(19)

Observe the pattern
X(2)-X(1) = 5-3 = 2 = 2^1
X(3)-X(2) = 9 - 5= 4 = 2^2.
X(4)-X(3) = 17-9 = 8 = 2^3.
So X(20)-X(19) must be 2^19.
Hence A is the answer.
Let me know if this helps :)
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by Anurag@Gurome » Tue Jun 05, 2012 7:52 pm
alex.gellatly wrote:If the sequence x<1>, x<2>,.....x<n>, is such that x<1> = 3 and x<n+1> = 2x<n>-1 for n less than or equal to 1, then x<20>-x<19>=

A. 2^19
B. 2^20
C 2^21
D 2^20 - 1
E 2^21 - 1

Thanks
x1 = 3
x2 = 2*3 - 1 = 5
x3 = 2*5 - 1 = 9
x4 = 2*9 - 1 = 17 and so on.
Generalizing, we get x(n) = 2^n + 1
So, x(20) = 2^20 + 1
x(19) = 2^19 + 1
x(20) - x(19) = 2^20 + 1 - 2^19 - 1 = 2^20 - 2^19 = 2^19(2 -1) = [spoiler]2^19[/spoiler]

The correct answer is A.
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by myselfhari » Thu Jun 07, 2012 2:43 am
yeah.. i too got the answer. it is very straight, however,there is a clause in the question stating, for all n less than or equal to 1 ..

How is this not considered while answering the question.?
because if this condition is true, then we will not be getting any value after x<2>.

So what about it ?
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by eagleeye » Thu Jun 07, 2012 4:20 am
Its a transcription error. I saw it when i solved it and figured that it must read "n is greater than or equal to 1."
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by alex.gellatly » Sun Jun 10, 2012 12:05 am
eagleeye wrote:Its a transcription error. I saw it when i solved it and figured that it must read "n is greater than or equal to 1."
I fixed the error, sorry about that. Thanks again for showing how to solve it.
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