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

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Intern
Joined: 04 Aug 2010
Posts: 13
If the sequence x1, x2, x3, ... xn, ... is such that x1 = 3  [#permalink]

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04 Aug 2010, 04:17
1
15
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Difficulty:

55% (hard)

Question Stats:

71% (02:12) correct 29% (02:24) wrong based on 199 sessions

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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
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Joined: 02 Sep 2009
Posts: 49273

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04 Aug 2010, 04:25
3
1
kwhitejr wrote:
Can anyone demonstrate the following?

If the sequence x1, x2, x3, ... xn, ... is such that x1 = 3 and xn+1 = 2xn - 1 for n = 1, then x20 - x19 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 $$x_{19}=2^{19}+1$$.

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

Hope it helps.
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Re: If the sequence x1, x2, x3, ... xn, ... is such that x1 = 3  [#permalink]

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14 Jul 2015, 18:29
x1 = 3
x2 = 2(3)-1 = 5
x3 = 2(5)-1 = 9
x4 = 2(9)-1 = 17

we can notice the general pattern in this series:
(x2-x1) = 2 = 2^1
(x3-x2)= 4 = 2^2
(x4-x3) = 8 = 2^3... and so on.

So, (x^20 - x^19) = 2^(19). Ans (A).
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Re: If the sequence x1, x2, x3, ... xn, ... is such that x1 = 3  [#permalink]

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24 Sep 2017, 15:13
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: If the sequence x1, x2, x3, ... xn, ... is such that x1 = 3 &nbs [#permalink] 24 Sep 2017, 15:13
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