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if x1, x2, x3......xn is such that x1 = 3 and xn+1 = 2.xn-1

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Manager
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if x1, x2, x3......xn is such that x1 = 3 and xn+1 = 2.xn-1 [#permalink]

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06 Sep 2004, 17:32
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if x1, x2, x3......xn is such that x1 = 3 and xn+1 = 2.xn-1 for all n >= 1 then x20-x19 =

1. 2^19

2. 2^20

3. 2^21

4. 2^20-1

5. 2^21-1

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Director
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06 Sep 2004, 17:41
X20=2X19-1
X19=2X18-1
-> X20-X19=2(X19-X18) .... recursive relation
or X20-X19=2^18(X2-X1)=2^19

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Manager
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06 Sep 2004, 18:00

i followed your steps all the way till the end but did not get how x2-x1 = 2

please let me know how you did that part
thanks

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06 Sep 2004, 18:21
srijay007 wrote:
X20=2X19-1
X19=2X18-1
-> X20-X19=2(X19-X18) .... recursive relation
or X20-X19=2^18(X2-X1)=2^19

X20-X19
=2(X19-X18)
=2*2(X18-X17)
.
.
.
=2^(19-2+1)*(X2-X1)

This is what I did

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Manager
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06 Sep 2004, 18:25
yep i got that part.. just wondering how you wetn from 2^18(x2-x1) to 2^19

i.e. x2-x1 = 2

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Director
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06 Sep 2004, 18:26
damit wrote:

i followed your steps all the way till the end but did not get how x2-x1 = 2

please let me know how you did that part
thanks

I solved x2=2*x1-1=5
x1 is given as 3
hence x2-x1=2

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06 Sep 2004, 19:25
What does the Xn-1 mean? Are they all in subcript, meaning,

if n =1, Xn-1 = 19

or is n the subscript meaning,

if n = 10, Xn-1 = X10 - 1

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06 Sep 2004, 19:42
n=19, Xn+1 = X20 = 2(X19-1) = 2(X18)
n=18, Xn+1 = X19 = 2(X18-1) = 2(X17)

So X20-X19 = 2(X18) - 2(X17) = 2(X18-X17)

X18 = 2(X16)
X17 = 2(X15)

So X20-X19 = 2^2(X16-X15)

So if we keep reducing it, we should eventually end up with

2^9(X2-X0)

I can't seem to get further than this.

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Director
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06 Sep 2004, 21:09
srijay007 wrote:
ywilfred wrote:
What does the Xn-1 mean? Are they all in subcript, meaning,

if n =1, Xn-1 = 19

or is n the subscript meaning,

if n = 10, Xn-1 = X10 - 1

Ya it's subscript
if n=10 Xn-1 means X9

No, I think they are not all subscripts. xn+1 = 2.xn-1 means xn+1 = (2.xn) - 1

My soln:
I solved till n=4, then generalised the sequence.
n1=3, n2=5, n3=9, n4=17..., (2^n+1)+1

Therefore: x20-x19
= 2(X19) - 1 - (X19)
= X19 - 1
= 2^19 + 1 -1
= 2^19

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Manager
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07 Sep 2004, 00:17
my 2 cents: ans - A- 2^19

x1 = 3
x2 = 5 - diff = 2
x3 = 8 - diff = 4
x4 = 17 - diff = 8
x5 = 33 - diff = 16
and so on

so the difference between consecutive terms follows the sequence
Xn - Xn-1 = 2^(n-1)
so X20 - X19 = 2^19

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07 Sep 2004, 00:17
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if x1, x2, x3......xn is such that x1 = 3 and xn+1 = 2.xn-1

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