Sequence Problem

[taneja.niks]

If the sequence x1, x2, x3, ..., xn, ... is such that x1 = 3 and xn+1= 2xn- 1 for n≥ 1, then x^20 - x^19 =

A. 2^19
B. 2^20
C. 2^21
D. 2^20 - 1
E. 2^21 - 1

[Anurag@Gurome]

If the sequence x1, x2, x3, ..., xn, ... is such that x1 = 3 and xn+1= 2xn- 1 for n≥ 1, then x^20 - x^19 =

A. 2^19
B. 2^20
C. 2^21
D. 2^20 - 1
E. 2^21 - 1

x(1) = 3
x(n+1) = 2x(n) - 1
x(2) = 2x(1) - 1 = 2*3 - 1 = 5
x(3) = 2x(2) - 1 = 2*5 - 1 = 9
x(4) = 2x(3) - 1 = 2*9 - 1 = 17
...
Generalizing, we get that x(n) = 2^n + 1

So, x(20) = 2^20 + 1 and x(19) = 2^19 + 1
Therefore, x^20 - x^19 = 2^20 + 1 - 2^19 - 1 = 2^19(2 - 1) = 2^19

The correct answer is A.

[taneja.niks]

Thanks for the reply I missed the 2^n relation.

[Anurag@Gurome]

Thanks for the reply I missed the 2^n relation.

Welcome!

[Ian Stewart]

If the sequence x1, x2, x3, ..., xn, ... is such that x1 = 3 and xn+1= 2xn- 1 for n≥ 1, then x^20 - x^19 =

A. 2^19
B. 2^20
C. 2^21
D. 2^20 - 1
E. 2^21 - 1

The notation you've used here is potentially confusing. It should read:

x_1 = 3
x_(n+1) = 2(x_n) - 1

The question then asks us to find x_20 - x_19 (and not x^20 - x^19).

As with any sequence question that gives a first term and a rule to find later terms, I'd work out the first few terms of the sequence:

x_1 = 3
x_2 = 2*3 - 1 = 5
x_3 = 2*5 - 1 = 9
x_4 = 17
x_5 = 33
etc

Now it is useful to look at the question to see what we are asked to find. The question asks us to find the *difference* between two consecutive terms, x_20 - x_19. While we could work out a pattern that lets us find x_20 and x_19 individually, the question doesn't ask for x_20 or x_19; that's more work than we need to do. Instead we can look for a pattern in the *differences* of consecutive terms:

x_2 - x_1 = 5 - 3 = 2 = 2^1
x_3 - x_2 = 9 - 5 = 4 = 2^2
x_4 - x_3 = 17 - 9 = 8 = 2^3
x_5 - x_4 = 33 - 17 = 16 = 2^4

which suggests that x_(n+1) - x_n = 2^n, and thus that x_20 - x_19 = 2^19.