Let ₂ (-2+ _ r=1 ¹⁰⁰ r 2^r )=a+ _c b where a, b, c are integers and a>b>c>0 . Then find the value of (a+b+c) .

Step-by-step solution

aligned S & =1 2^1+2 2^2+3 2^3+ +100 2¹⁰⁰ 2 S & = 1 2^2+2 2^3+3 2^4+ +99 2¹⁰⁰+100 2¹⁰¹ aligned Sub aligned & -S= (2+2^2+2^3+ +2¹⁰⁰ )-100 2¹⁰¹ & -S=2 (2¹⁰⁰-1 )-100 2¹⁰¹ & S=100 2¹⁰¹-2¹⁰¹+2 & S=99 2¹⁰¹+2 & S-2=99 2¹⁰¹ & ₂ (99 2¹⁰¹ )=101+ ₂ 99 a+ _c b & a=101, b=99, c=2 & aligned