Given x_n, how to find x_(n+1)?

Question

Given a sequence x_n, how do you find x_(n+1)? For example, I have two sequences a_n = ((1*3*5*...*(2n-1))/(2*4*6*...*(2n)) and b_n = ((2n+1)!/10^n), and respectively a_n+1 = ((1*3*5*...*(2n-1)(2n+1))/(2*4*6*...*(2n)(2n+2)) and b_n+1 = ((2n+1)!(2n+3)/(10^n)*10). I don't seem to notice what is the pattern. Could someone explain it?

Answer 1

A sequence is represented by a₀, a₁, a₂​​​​​​​, a₃​​​​​, ..., a_n, where a_n=f(n) is a possible function for determining a_n generally. But this is only one of a number of ways of defining a_n. For example, a Fibonacci series defines a_n=a_n-2+a_n-1 and a_n cannot be defined (exactly) by using a function of n, although there is a formula for an approximation of a_n.

But if a_n is given as a function of n, then a_n+1 can easily be found by substitution.

We​​​​​ can also find a_n+1-a_n and thereby establish a relationship between consecutive terms. A simple example is, given a_n=(2n+1)/(2n-1), the series would be:

-1, 3, 5/3, 7/5, 9/7, ...

a_n+1=(2(n+1)+1)/(2(n+1)-1)=(2n+3)/(2n+1);

a_n+1-a_n=(2n+3)/(2n+1)-(2n+1)/(2n-1)=[(2n+3)(2n-1)-(2n+1)²]/(4n²-1),

a_n+1-a_n=(4n²+4n-3-4n²-4n-1)/(4n²-1)=-4/(4n²-1), so:

a_n+1=a_n-4/(4n²-1).

Now take one of your examples. a_n=(1)(3)(5)...(2n+1)/[(2)(4)(6)...(2n)]. 

a_n+1=(1)(3)(5)...(2n+3)/[(2)(4)(6)...(2n+2)].

a_n+1-a_n=(1)(3)(5)...(2n+3)/[(2)(4)(6)...(2n+2)]-(1)(3)(5)...(2n+1)/[(2)(4)(6)...(2n)],

a_n+1=a_n+[(2n+3)/(2n+2)]{(1)(3)(5)...(2n+1)/[(2)(4)(6)...(2n)]}. Although it's complicated, there is a clear function of n which links successive terms of the series. This series could also be expressed:

Given a₀=1, a_n+1=(2n+3)/(2n+2)a_n, generating the series a_n=1, 3/2, 15/8, 105/48=35/16, ... Note that this is different from a_n+1=a_n+f(n), which is a sum, not a product.

I think the problem you have observed is, not how to calculate x_n+1, but how to calculate a_n as f(n), a function of n. If and when you can do so, then you will be able to calculate another function of n, g(n), which relates successive terms in the series.