Find using Progression!!

$Given\quad is\\ Say\quad M\quad =\quad { 2 }^{ \frac { 1 }{ 4 } }\quad *\quad { 4 }^{ \frac { 1 }{ 8 } }\quad *\quad { 8 }^{ \frac { 1 }{ 16 } }\quad *\quad { 16 }^{ \frac { 1 }{ 32 } }\quad ....\\ M\quad =\quad { 2 }^{ (\frac { 1 }{ 4 } +\frac { 2 }{ 8 } +\frac { 3 }{ 16 } +\frac { 4 }{ 32 } ...) }\\ Let\quad S\quad =\quad \frac { 1 }{ 4 } +\frac { 2 }{ 8 } +\frac { 3 }{ 16 } +\frac { 4 }{ 32 } ...\quad \quad \quad \quad (1)\\ 0.5\quad S\quad =\quad \quad \quad \quad \frac { 1 }{ 8 } +\frac { 2 }{ 16 } +\frac { 3 }{ 32 } +\frac { 4 }{ 64 } ...\quad (2)\\ By\quad (1)\quad -\quad (2)\\ 0.5\quad S\quad =\quad \frac { 1 }{ 4 } +\frac { 1 }{ 8 } +\frac { 1 }{ 16 } +\frac { 1 }{ 32 } ...\\ Which\quad is\quad geometric\quad series\\ \therefore \quad S\quad =\quad \frac { 1 }{ 2 } (\frac { \frac { 1 }{ 4 } }{ (1\quad -\quad \frac { 1 }{ 2 } ) } )\quad =\quad 1.\\ M\quad =\quad { 2 }^{ 1 }\quad =\quad 2.\\ This\quad can\quad be\quad also\quad solve\quad using\quad Arithmatico-geometric\quad series.$