If the sequence x1, x2, x3, ... xn, ... is such that x1 = 3 : GMAT Problem Solving (PS)
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# If the sequence x1, x2, x3, ... xn, ... is such that x1 = 3

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If the sequence x1, x2, x3, ... xn, ... is such that x1 = 3 [#permalink]

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25 Mar 2006, 13:50
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If the sequence $$x_1$$, $$x_2$$, $$x_3$$, ..., $$x_n$$, ... is such that $$x_1 = 3$$ and $$x_{n+1} = 2x_n - 1$$ for $$n\geq1$$, then $$x_{20} - x_{19}$$ equals which of the following?

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

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-the-sequence-x1-x2-x3-xn-is-such-that-x1-98536.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 24 May 2014, 05:50, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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25 Mar 2006, 15:22
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Well the answer seems to be 2^19 and A (although your A states 219).
Just write down the first elements and compute the differences.

x1=3
x2=5
x3=9
x4=17

So
x2=x1+2^1
x3=x2+2^2
x4=x3+2^3
... and
x20=x19+2^19, so

x20-x19=2^19.
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24 May 2014, 05:33
ccax wrote:
Well the answer seems to be 2^19 and A (although your A states 219).
Just write down the first elements and compute the differences.

x1=3
x2=5
x3=9
x4=17

So
x2=x1+2^1
x3=x2+2^2
x4=x3+2^3
... and
x20=x19+2^19, so

x20-x19=2^19.

sorry how did you get x2 =5 and so on? I'm not even understanding the question....x (n+1)=2x(n)-1......isnt this the questions? then how does everything goes into powers of two? Please help!
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24 May 2014, 05:52
usre123 wrote:
ccax wrote:
Well the answer seems to be 2^19 and A (although your A states 219).
Just write down the first elements and compute the differences.

x1=3
x2=5
x3=9
x4=17

So
x2=x1+2^1
x3=x2+2^2
x4=x3+2^3
... and
x20=x19+2^19, so

x20-x19=2^19.

sorry how did you get x2 =5 and so on? I'm not even understanding the question....x (n+1)=2x(n)-1......isnt this the questions? then how does everything goes into powers of two? Please help!

If the sequence $$x_1$$, $$x_2$$, $$x_3$$, ..., $$x_n$$, ... is such that $$x_1 = 3$$ and $$x_{n+1} = 2x_n - 1$$ for $$n\geq1$$, then $$x_{20} - x_{19}$$ equals which of the following?

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

We have the sequence $$x_1$$, $$x_2$$, $$x_3$$, …, $$x_n,$$… $$x_1=3$$ and $$x_{n+1}=2x_n - 1$$ for $$n\geq1$$.

If you notice there is a specific pattern in it:

$$x_1=3=2^1+1$$
$$x_2=2x_1-1=5=2^2+1$$
$$x_3=2x_2-1=9=2^3+1$$
...
$$x_n=2^n+1$$

So, $$x_{20}=2^{20}+1$$ and $$x_{19}=2^{19}+1$$.

$$x_{20}-x_{19}=2^{20}+1-2^{19}-1=2^{20}-2^{19}=2^{19}$$

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-the-sequence-x1-x2-x3-xn-is-such-that-x1-98536.html
_________________
Re: PS-sequence   [#permalink] 24 May 2014, 05:52
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