Sequences

zagcollins 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.2^21
D.2^20-1
E.2^21-1

With a sequence question, it is almost always a good idea to write out the first few terms. Here, we know x(1) = 3. The other terms are defined by the rule:

x(n+1) = 2x(n) - 1

The (n+1)th term is the term that follows the (n)th term, so you can read the above definition as follows: "to find the next term, multiply the current term by 2, then subtract 1". In that way, we can quickly find a few terms:

x(1) = 3
x(2) = 5
x(3) = 9
x(4) = 17
x(5) = 33

What we don't want to do is find the value of x(20) and of x(19)- that could take all day. There are many ways to answer the question, but normally sequence questions test your ability to recognize a pattern. The question asks about the difference between consecutive terms: it asks for x(20) - x(19). We can look at the difference between the terms we wrote down above, and see if a pattern emerges:

x(2) - x(1) = 2 = 2^1
x(3) - x(2) = 4 = 2^2
x(4) - x(3) = 8 = 2^3
x(5) - x(4) = 16 = 2^4
and from this it seems likely that...
x(20) - x(19) = 2^19

That's not a rigorous mathematical proof, but for a GMAT sequence question, it will be a reliable method almost all of the time.

There are many ways to prove this rigorously; one is by 'mathematical induction':

x(1) = 3 = 2^1 + 1

If x(n) = 2^n + 1, then
x(n+1) = 2x(n) - 1 = 2*(2^n + 1) - 1 = 2^(n+1) + 1

So for all n, x(n) = 2^(n) + 1

Thus,
x(20) - x(19) = (2^(20) + 1) - (2^(19) + 1) = 2^20 - 2^19 = 2^19