netigen wrote:
If the sequence x1, x2, x3, …, xn, … is such that x1 = 3 and x(n+1) = 2(xn) – 1 for n = 1, then x20 – x19 =
The part I've bolded: I'm sure you mean
for n>1.
With any sequence question, I think it's a good idea to write down the first three or four terms, at least. It's normally very fast to do, and if a pattern emerges, you'll be more likely to see it. Here we have:
x_1 = 3
x_2 = 5
x_3 = 9
x_4 = 17
The question asks about the difference of consecutive terms, so look there for a pattern:
x_2 - x_1 = 5 - 3 = 2^1
x_3 - x_2 = 9 - 5 = 2^2
x_4 - x_3 = 17 - 9 = 2^3
From there, it's reasonable to predict that:
x_(n+1) - x_n = 2^n
x_20 - x_19 = 2^19
That's not what a mathematician would consider an airtight proof- we've just recognized a pattern, and guessed it will continue forever. But it does give the right answer, and normally will on the GMAT.
We can certainly do this more rigourously. Again, if we write the first terms down:
x_1 = 3
x_2 = 5
x_3 = 9
x_4 = 17
we notice that, for 1 <= n <= 4, x_n = 2^n + 1. Will all the terms be in this form?
Using the definition of x_(n+1), if x_n = 2^n + 1, then:
x_(n+1) = 2x_n - 1 = 2(2^n + 1) -1 = 2^(n+1) + 1
So if one term is in this form, every term that follows must also be in this form- indeed for every term, x_n = 2^n + 1.
Then
x_20 - x_19 = 2^20 + 1 - (2^19 + 1) = 2^20 - 2^19 = 2^19.
There are quite a few other ways to get to the answer here, though I'm not seeing a '15 second' solution. There may well be one, however.
