Hard progression: x0=1, x(n+1)=2x(n)+1, S(n)=x0+x1+...x(n). Find S(2018)

[wolly]

I have the following problem:
x0=1
x(n+1)=2x(n)+1
S(n)=x0+x1+...x(n)
Find S(2018)
How do I find it?
I know that the terms are:x0=1,x1=3,x2=7,x3=15 and that the difference of the terms is 2,4,8 but how does that help me to find the formula for the sequence x(n)?

 

[Deleted member 4993]

I have the following problem:
x0=1
x(n+1)=2x(n)+1
S(n)=x0+x1+...x(n)
Find S(2018)
How do I find it?
I know that the terms are:x0=1,x1=3,x2=7,x3=15 and that the difference of the terms is 2,4,8 but how does that help me to find the formula for the sequence x(n)?

x(n) + x(n+1) = 3x(n) + 1

x(n-1) + x(n) + x(n+1) = x(n-1) + 3x(n) + 1 = 7x(n-1) + 4

continue and see if any pattern emerges....

 

[wolly]

x(n) + x(n+1) = 3x(n) + 1

x(n-1) + x(n) + x(n+1) = x(n-1) + 3x(n) + 1 = 7x(n-1) + 4

continue and see if any pattern emerges....

I just found out that x(n)=2^(n+1)-1.I'm not sure how your expression helps me.Now the question was how do I find the S(2018)?Also,did 2^(n+1)-1 come from the difference of terms?Example: x1-x0=3-1=2
x2-x1=7-3=4
x3-x2=15-7=8
... And so on

 

[lev888]

I just found out that x(n)=2^(n+1)-1.I'm not sure how your expression helps me.Now the question was how do I find the S(2018)?Also,did 2^(n+1)-1 come from the difference of terms?Example: x1-x0=3-1=2
x2-x1=7-3=4
x3-x2=15-7=8
... And so on

If x_n = 2ⁿ⁺¹ - 1 consider summing up 2ⁿ⁺¹ and "-1" terms separately.

 

[Denis]

Have a Sloanic peek:

http://oeis.org/search?q=1,3,7,15,31,63&sort=&language=english&go=Search

Don't forget to "donate".

 

[pka]

I have the following problem:
x0=1
x(n+1)=2x(n)+1
S(n)=x0+x1+...x(n)
Find S(2018)
How do I find it?
I know that the terms are:x0=1,x1=3,x2=7,x3=15 and that the difference of the terms is 2,4,8 but how does that help me to find the formula for the sequence x(n)?

You might find THIS useful.