Series

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

In sequence questions, it's a very good idea to write down the first few terms, to see if a pattern emerges:

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.