Question

determinati n€N astfel incat fractia 1+2+2 la puterea 2+...+2 la puterea 100 / 2 la puterea n sa fie: A)subunitara, B) supraunitara

Answer 1

1+2+2^2+2^3+...+2^100=S  l  x2

2+2^2+2^3+2^4+...+2^100+2^101=2S  l  +1

1+2+2^2+2^3+...+2^100+2^101=2S+1

(1+2+2^2+2^3+...+2^100)+2^101=2S+1

S+2^101=2S+1

S=2^101 -1

a)
(2^101 -1)/2^n >1

2^101 -1>2^n

n<=100

b)
(2^101 -1)/2^n <1

2^101 -1<2^n

n>=101