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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

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.

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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