What is the sum of

2 raised to 0 ---------------------------------------2 raised to 2008

let s=2⁰+2¹+--------------------2²⁰⁰⁸

let s₁=2¹+2²⁺---------------2²⁰⁰⁸

s₁ is a geometric progression with a=2,r=2 & n=2008

so, s₁=a(rⁿ-1)/r-1

s₁=2(2²⁰⁰⁸-1)/2-1 = 2²⁰⁰⁹-2 

now, s=2⁰+ s₁=1+2²⁰⁰⁹-2=2²⁰⁰⁹-1.