x20 - x19 = 2*x19 - 1 - 2*x18 + 1 = 2(x19 - x18) =
= 2(2*x18 - 1 - 2*x17 + 1) = 2^2(x18 - x17) =
= .... = 2^18(x2 - x1).
X2 = 2*x1 - 1 = 5, meaning that x2 - x1 = 5 - 3 = 2.
This makes the answer you're looking for 2^18 * 2 = 2^19.
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In sequence questions, it's a very good idea to write down the first few terms, to see if a pattern emerges:ketkoag wrote:If the sequence x1, x2, x3, …, xn, … is such that x1 = 3 and xn+1 = 2xn – 1 for n ≥ 1, then x20 – x19 =
A. 2^19
B. 2^20
C. 2^20 - 1
E. 2^21 - 1
OA: A
a_1 = 3
a_2 = 5
a_3 = 9
a_4 = 17
The question asks about a_20 - a_19 (that is, it asks about the difference between consecutive terms), so let's do the same for the terms above:
a_2 - a_1 = 2 = 2^1
a_3 - a_2 = 4 = 2^2
a_4 - a_3 = 8 = 2^3
and if we trust that this pattern will continue, we would conclude that
a_(n+1) - a_n = 2^n
so a_20 - a_19 should be equal to 2^19.
Of course, that's not a mathematically rigorous proof, but we don't need a rigorous proof to answer a GMAT question. Many sequence questions are testing whether you can recognize and extend a pattern, so the approach above can be used reliably in several questions.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com
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