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

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

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28 Nov 2005, 07:10
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If the sequence x1, x2, x3, â€¦, xn, â€¦ is such that x1 = 3 and xn+1 = 2xn â€“ 1 for n ≥ 1, then x20 â€“ x19 =

A. 219
B. 220
C. 221
D. 220 - 1
E. 221 - 1
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Tanmoi.

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28 Nov 2005, 16:21
X(n+1) = 2X(n) - 1

X(1) = P
X(2) = 2P - 1
X(3) = 2(2P - 1) - 1 = 4P - 2 - 1
X(4) = 2(4P - 2 - 1 ) - 1 = 8P - 4 - 2 - 1
X(5) = 2(8P - 4 - 2 - 1) - 1 = 16P - 8 - 4 - 2 - 1

Thus, X(n) = 2^(n-1)P - 2^(n-1) + 1

X(19) = 2^18 * P - 2^18 + 1
= 2^18(P - 1) + 1
= 2^19 + 1
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28 Nov 2005, 16:45
Tanmoi wrote:
If the sequence x1, x2, x3, â€¦, xn, â€¦ is such that x1 = 3 and xn+1 = 2xn â€“ 1 for n ≥ 1, then x20 â€“ x19 =

A. 219
B. 220
C. 221
D. 220 - 1
E. 221 - 1

can you double-check the bold part?? ...xn+1 is xn + 1?? or the (n+1)th term???

the answer choices, are they 2^ 19, 2^20,...?
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28 Nov 2005, 17:28

A = 219, but D is also 219 (220-1), B is 220, but E is also 220 (221-1)

Something wrong?? Or there's some subscript that we're not aware of
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29 Nov 2005, 03:56
Tanmoi wrote:
If the sequence x1, x2, x3, â€¦, xn, â€¦ is such that x1 = 3 and xn+1 = 2xn â€“ 1 for n ≥ 1, then x20 â€“ x19 =

A. 219
B. 220
C. 221
D. 220 - 1
E. 221 - 1

I think the answer choices should be 2 to the powers. Here is my working.
X1 =3,
X2=5
X3=9
X4-17, we are having a pattern here. It is Xn = (2^n)-1
So X20 = (2^20)-1
X19 = (2^19) -1
X20 -X19 = (2^20-1)-(2^19-1)
= 2^20 - 2^19
= 2^19(2-1)
= 2^19
A.
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29 Nov 2005, 06:48
I get 2^19 as well as the difference between two consecutive terms can be expressed as 2 raised to power of the lower term.

x1 = 3
x2 = 5 (x2-x1 = 2 or 2^1)
x3 = 9 (x3-x2 = 4 or 2^2)
x4 = 17 (x4-x3 = 8 or 2^3)

So x12-x11 = 2^11
And x20-x19 = 2^19

Furthermore, the difference between non-consecutive terms can be expressed by summing the differences of the consecutive terms in between them, that would have been an interesting question.

For example, x5 - x2 = 2^2+2^3+2^4 = 28 which is indeed 33-5.
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29 Nov 2005, 20:36
could somebody post A detailed explanation, please?
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29 Nov 2005, 20:36
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