Question

Fie M, mijlocul segmentului[AB] ,M2 mijlocul segmentului [AM1],...,Mn mijlocul segmentului [AMn-1].Aflati numarul natural n, stiind ca AMn=1 si AMn+AMn-1+...+AM1=127​

Answer 1

A----M4-----M3----------M2---------------------M1---B

AM1 = 2AM2

AM2 = 2AM3

AM3 = 2AM4

.....

AM(n-1) = 2AMn

=> AM(n-1) = 2×1 = 2 = 2¹

=> AM(n-2) = 2×2 = 4 = 2²

=> AM(n-3) = 4×2 = 8 = 2³

.....

=> AM2 = AM(n-(n-2)) = 2^(n-2)

=> AM1 = AM(n-(n-1)) = 2^(n-1)

=> AMn + AM(n-1) +...+AM1 = 127

=> 1+2¹+2²+2³+...+2^(n-1) = 127

=> 1+2(2^(n-1)-1)/(2-1) = 127

=> 1+2^n -2 = 127

=> 2^n = 128

=> n = 7